EDITED BY 

ALFRED L. HALL-QUEST 



SUPERVISED STUDY IN MATHEMATICS 
AND SCIENCE 



•?$&&■ 



THE MACMILLAN COMPANY 

NEW YORK • BOSTON • CHICAGO • DALLAS 
ATLANTA • SAN FRANCISCO 

MACMILLAN & CO., Limited 

LONDON • BOMBAY • CALCUTTA 
MELBOURNE 

THE MACMILLAN CO. OF CANADA, Ltd. 

TORONTO 



SUPERVISED STUDY 

IN 

MATHEMATICS AND SCIENCE 



BY 

S. CLAYTON SUMNER, M.A. 

SUPERVISING PRINCIPAL, PALMYRA, N. Y. 
(Formerly at Canton, N. Y.) 



THE MACMILLAN COMPANY 

1922 

AH rights reserved 



PRINTED IN THE UNITED STATES OF AMERICA 



Q 3%* 



Copyright, 1922, 
By THE MACMILLAN COMPANY. 



Set up and electrotyped. Published November, 1922. 



NorfoflotJ press 

J. S. Cushing Co. — Berwick & Smith Co. 

Norwood, Mass., U.S.A. 



NOV 16 '22 



C1A690215 
I 






Zo tbe /IBemotE of 
MY FATHER 

WHO BELIEVED AN EDUCATION WAS THE 

RICHEST HERITAGE A PARENT COULD 

BEQUEATH TO HIS CHILDREN 



PREFACE 

In attempting a book on supervised study which will cover 
even approximately the subjects of mathematics and science, 
it is impossible to do more than give suggestive lessons. This, 
therefore, has been my plan — to give only one or two typi- 
cal outlines of a topic or subject, but to leave an intimation 
of its application whenever or wherever the teacher may elect. 
Thus, only one Red Letter Day lesson is presented in Algebra, 
but the teacher will undoubtedly desire to use many such 
plans during the year. The material in the lessons men- 
tioned may be suggestive for the planning of others. 

I have not tried to add another learned book in pedagogy 
to the many already on the market. It has rather been my 
aim to write a book that may be of explicit and direct value 
to the teacher or principal who is daily striving to teach his 
children how to study and how to learn, I have tried to write 
it in simple language, so that the reader may get the meat, 
if there be any, without too much stuffing. 

It is needless to say that I am a firm believer in supervised 
study. It has done much for our children ; I am sure it will 
do more as we progress in the proficiency of its administra- 
tion. It is not a panacea for all pedagogical ills, but it is 
valuable for what it claims to be, and it holds great promise 
for the future. 

I am greatly indebted to Professor Alfred L. Hall-Quest 
of the University of Cincinnati, who, as editor of this series, 
has not only made it possible for this volume to be, but who, 



viii Preface 

through the reading and criticism of the manuscript and 
through innumerable other suggestions, has been of ines- 
timable help to me. 

Deep appreciation is also here expressed to Professor 
Charles M. Rebert of St. Lawrence University, for valuable 
suggestions and advice ; to Mr. A. E. Breece of the Hughes 
High School, Cincinnati, Ohio, who made a very careful and 
valuable critical review of the manuscript as relating to mathe- 
matics ; to my teachers at Canton, N. Y., who made it possi- 
ble to actually try out many of the lessons ; and to my wife, 
for her constant counsel and encouragement. 

In addition, I wish to acknowledge my thanks for the cour- 
tesy of The Macmillan Company, the American Book Com- 
pany and the Charles E. Merrill Company for permission 
to quote more or less extensively from their publications. 

S. Clayton Sumner. 

Palmyra, N. Y. 
January 31, 1922. 



TABLE OF CONTENTS 

PAGE 

Introduction. Supervised Study a Moral Imperative . . The Editor xiii 

PART ONE. MATHEMATICS 

Chapter One. Management of the Supervised Study Period in Math- 
ematics 3 

First Section. Algebra (Elementary) 

Chapter Two. Divisions of Elementary Algebra; Units of Instruc- 
tion and Units of Recitation. A Time Table . . 20 

ILLUSTRATIVE LESSONS 

LESSON 

I. The Inspirational Preview 26 

II. Introduction. Unit of Instruction I. A Lesson in Correlation 34 

EEL Introduction {Continued). A How to Study Lesson . . 43 

IV. Introduction [Continued). An Inductive and How to Study 

Lesson 50 

V. Addition. Unit of Instruction IH. An Inductive Lesson: 

Addition of Monomials 59 

VI. Addition (Continued). An Inductive Lesson: Addition of 

Polynomials 67 

VQ. Simple Equations. Unit of Instruction X. An Expository 

and How to Study Lesson : The Equation and Problems . 73 

VHI. Factoring. Unit of Instruction VII. A Socialized Lesson . 80 



x Table of Contents 

LESSON PAGE 

IX. Fractions. Unit of Instruction IX. A Deductive and How 

to Study Lesson 82 

X. A Red Letter Day Lesson 85 

XI. Radicals. Unit of Instruction XIV. A Socialized Lesson . 88 

XII. Quadratic Equations. Unit of Instruction XV. An Exposi- 
tory and How to Study Lesson . . . . . .92 

XIII. An Examination 95 

Second Section. Plane Geometry 

Chapter Three. Divisions of Plane Geometry; Units of Instruction 

and Units of Recitation 105 

LESSON 

I. The Inspirational Preview 106 

II. Rectilinear Figures. Unit of Instruction II. A Deductive 

and How to Study Lesson : Vertical Angles . . .110 

III. Rectilinear Figures {Continued). A Deductive Lesson: Tri- 

angles . . . .117 

IV. Rectilinear Figures {Continued). A How to Study Lesson: 

Originals 123 

V. Rectilinear Figures {Continued). A Deductive Lesson : Orig- 
inals 129 

VI. A Socialized Review : Book I 134 

VII. An Exhibition or a Red Letter Day Lesson . . . .135 



Third Section. Advanced Mathematics 

Chapter Four. Special Methods of Supervised Study in Higher Math- 
ematics , 141 



Table of Contents xi 

PART TWO. SCIENCE 

PAGE 

Chapter Five. The Management of the Supervised Study Period in 

Science ■>..-• 149 

Fourth Section. Biology 

Chapter Six. Divisions of Biology ; Units of Instruction and Units of 

Recitation. A Time Table 155 

A. — BOTANY 

LESSON 

I. The Inspirational Preview 156 

II. Introductory Topics. Unit of Instruction I. A How to Study 

Lesson: Preliminary Experiments 161 

III. Introductory Topics {Continued). An Inductive Lesson. 

Problem: No Two Plants Are Alike 165 

IV. Introductory Topics {Continued). An Inductive Lesson. 

Problem : Struggle for Existence 169 

V. Seeds and Seedlings. Unit of Instruction II. A How to Study 

Lesson: Seeds and Their Germination . . . .171 

VI. Seeds and Seedlings {Continued). A Deductive Lesson: Lab- 
oratory Experiments . . . . . . . .175 

VII. Seeds and Seedlings {Continued). A Socialized Lesson: A 

Field Trip 179 

B. — ZOOLOGY 

VIII. Insects. Unit of Instruction IX. A How to Study Lesson : 

The Grasshopper 181 

IX. Insects {Continued). A Laboratory Lesson: The Grasshop- 
per 187 

X. Insects {Continued). A Correlation and Research Lesson . 189 

XI. Insects {Continued). A Socialized Lesson . . . .192 

XII. A Red Letter Day Lesson 195 



xii Table of Contents 

C. — PHYSIOLOGY 

LESSON PAGE 

XIII. Bones and Muscles. Unit of Instruction XVI. A How to 

Study Lesson : Muscles 197 

XIV. Muscles {Continued). A Laboratory Lesson Using Micro- 

scopic Slides 200 

XV. Muscles {Continued). A Deductive Lesson. Problem: How 

Muscular Activity Is an Aid to Good Health . . . 202 

XVI. Muscles {Continued). A Lesson in Correlation . . . 204 

XVII. An Examination Lesson 206 

Fifth Section. Physics 
Chapter Seven. Further Lessons in Science 213 

LESSON 

I. Fluids. Unit of Instruction. An Expository and How to 

Study Lesson 214 

II. Fluids {Continued). A Laboratory Lesson . . . ,219 

III. Fluids {Continued). A How to Study Lesson : Problems . 223 

IV. Red Letter Day Lessons 225 

Bibliography 229 

Index 233 



INTRODUCTION BY THE EDITOR 

SUPERVISED STUDY A MORAL IMPERATIVE 

Millions of words have been written about education. 
Theories have abounded and still are fertile. The visitor at 
educational conventions, especially in the department of 
school superintendents, is impressed, however, with the rapid 
multiplication of devices for visualizing educational prac- 
tice. A rich variety of moving picture machines and already 
voluminous catalogues of educational films witness to the 
dawn of a new era in the technic of teaching. Later we shall 
no doubt find boards of censors passing upon these films — 
boards composed of theorists, critic teachers, educational 
scientists, et al., — but at present the field is open for all. 
Doubtless many teachers will find in this form of visual edu- 
cation an opportunity for enlargement of income as well as 
for the demonstration of teaching skill. 

Increasing Emphasis on Demonstration. Demonstration 
and description are rapidly coming to the front in discussions 
of methods of teaching. One carefully prepared and suc- 
cessfully performed demonstration is of more value than 
many verbal descriptions, however clear these may be. A 
series of vivid verbal descriptions makes definite and con- 
crete a volume of abstractions and theorizings on educational 
practice. Theory is important ; it must not be discounted ; 
but here, as elsewhere, an illustration turns on light and makes 

xiii 



xiv Introduction by the Editor 

objective and easily understood the necessarily vaguer dis- 
cussions of abstract theory. 

This series of volumes on Supervised Study attempts to 
visualize one form of study supervision. Each book is writ- 
ten by a teacher who has had considerable experience in this 
type of work. The emphasis in each discussion is to make 
concrete in as detailed description as possible, what the author 
has actually done in his own classroom. Very briefly each 
author states the theory underlying his practice, but beyond 
this brief statement he refrains from a discussion of principles. 
Teachers desire to see how theory is applied. One cannot be 
too clear and too definite in describing the mode of procedure 
in supervision. 

At Present No Generally Accepted Meaning of Supervised 
Study. In answer to those who believe that Supervised Study 
as described in this and other volumes of the series is different 
from the general understanding of the term, it should be em- 
phasized that at present there is no generally accepted form 
of Supervised Study. It is the conviction of the editor of 
this series that a standardized form is undesirable. The 
main objective is teaching children — all children — how 
to study and guiding them while they apply the principles of 
correct studying. It is of comparatively little importance 
how this is done, providing it is done effectively. If the 
teacher makes this type of teaching superlatively significant, 
it follows that the management of the class and the method 
of presenting subject matter will change accordingly. But 
each teacher must be the final judge of how to adapt this new 
point of view to local needs. 

The Imperative Need of Preventing Failures in School. It 
should be said, however, that any plan which seeks to prevent 



Introduction by the Editor xv 

failures and which aims to train all pupils to study as effec- 
tively as native ability permits is superior to plans that simply 
correct improper methods of work and that are concerned 
only with the retarded pupils. If school work is limited to 
the assigning and hearing of lessons, only a few — the highly 
endowed — will permanently profit by such experience. 
There are well-meaning people who sincerely believe that the 
school is the place for eHminating society's mentally unfit, 
and that the surest way of such elimination is to assign les- 
sons, long and hard. Those who can will ; those who cannot 
will not. Those who will and can are the fit ! 

Some there are who learn to swim by the "sink or swim" 
method ; they are destined, forsooth, to be swimmers if they 
do not sink. But how many of you who read these pages 
learned to swim by this fatalistic method? There are children 
who early judge themselves incapable of school work. No- 
body cares ! They either can or cannot study. By means 
of the hard, soulless machinery of assigning and hearing les- 
sons they are cast out. We call this a safe test and out they 
go labeled mentally weak, unfit to partake in a world of thrill- 
ing knowledge, unfit to climb to altitudes of self-revelation 
and social worth. If, however, they could have been taught 
how to use their minds, how to partake in the feast of knowl- 
edge, who knows but that many of them would have found 
a new meaning of their destiny ! 

Supervised Study Is Not Only an Intellectual Necessity; It Is a 
Moral Imperative. As teachers it is our plain duty to teach 
children how to study. The whole class period must be con- 
ducted in this spirit. The specific aim of every class period 
must be to so direct the pupils that their grasp of the new 
work is adequate for independent application. The teacher 



xvi Introduction by the Editor 

is preeminently a director of study and not primarily a dis- 
penser of subject matter. 

The Point of View in This Volume. The author of this 
volume is convinced of the effectiveness of Supervised Study. 
He and his teachers have tried it long enough to know its 
advantages. The subjects of mathematics and science are 
especially favorable to this method. A comprehensive view of 
the courses in the high school is given in a series of typical 
lessons describing in great detail how children may be directed 
in beginning, continuing, and reviewing their study of par- 
ticular units of subject matter. The author is well aware of 
the movement for reorganization of courses especially in ninth 
grade mathematics, but inasmuch as such revision is not 
likely to be possible in all schools for some time to come the 
usual division of courses is considered in this volume. It is 
believed also that even where general mathematics is taught, 
not a few pupils will elect additional special courses in the 
field of mathematics. Inasmuch as general science is at 
present little more than a combination of various special 
sciences the separate treatment used in this volume seems 
preferable. It is hoped that general science will evolve in- 
creasingly along the lines of natural correlations through 
which the pupil will be able to understand the intimate 
relationships that exist among the phenomena of nature. 



PART ONE 
MATHEMATICS 



SUPERVISED STUDY IN 
MATHEMATICS AND SCIENCE 

CHAPTER ONE 

THE MANAGEMENT OF THE SUPERVISED STUDY PERIOD 
IN MATHEMATICS 

Causes of Failures in Mathematics. — There are a number 
of contributory causes which, together or separately, might 
account for the high mortality in mathematics classes. That 
it is high is so commonly accepted among the profession, that 
a large percentage of failures has almost come to be an es- 
tablished expectation. In eleven high schools near Chicago, 
the percentage of failures in algebra and geometry was found 
to be greater than in any other subject. 1 In the report of 
the New York State Education Department on statistics 
for Regents Academic Examinations, the failures in mathe- 
matics for the past five years have been between thirty- 
three per cent and forty per cent. 2 The nearest competitors 
for scholastic dishonors are the commercial subjects which 
are largely mathematical in content. 

The causes of these failures are psychological, pedagogical, 
and physical. Psychologically, mathematics has been by 

1 School Review, June, 1913, p. 415. 

2 Annual Report of the State Department of Education (10th to 14th in- 
clusive), New York State. 

3 



4 Supervised Study in Mathematics and Science 

almost common opinion accorded the position of being the 
hardest subject in the school curriculum. This estimate of 
the subject, persisted in by pupils, teachers, and the laity, 
has inevitably resulted in a state of mind that predetermines 
a large percentage of failures. Until we teachers succeed in 
dispelling this opinion, pupils in many instances will expect 
to fail, and they will fail. There is no sane reason why mathe- 
matics should be so considered, and with the new vision of 
teaching the subject and with the readjustment of the course 
of study, combined with its scientific treatment (which will 
emphasize the functional and practical side instead of the 
formal aspect), this view of the severity of mathematics 
doubtless will gradually disappear. 

Mathematics Taught with Deliberate Unattractiveness. — It 
is repeating a platitude to refer to the fact that mathematics 
has been very poorly taught in the public school. There has 
been no serious lack of scholarship and of emphasis on the 
acquirement of knowledge of subject matter, but this very em- 
phasis has tended toward the serious neglect of training pupils 
to apply mathematical rules and formulas to practical reason- 
ing. Too much emphasis has been laid on the formal examina- 
tion, the " spectacular " effects according to Schultze. 1 Too 4 
much is attempted in the time allotted, with insufficient as- 
similation of the matter studied. Pupils are not taught 
how to study mathematics. They are only drilled on abstract 
formulas. The result is an overdeveloped memory and 
undeveloped powers of reasoning. 

Because of the above noted unsound pedagogical methods, 
with the resulting formal examinations, and because the 

1 Arthur Schultze, "The Teaching of Mathematics in Secondary Schools"; 
The Macmillan Company, 191 2. 



Supervised Study Period in Mathematics 5 

pupils are graded chiefly on mechanical ability, their prog- 
ress in mathematics can be determined to a highly refined 
nicety. They have failed to " do " a certain number of prob- 
lems. Ergo, they are just that much deficient in ability and 
improvement. There is no leeway for difference of opinion, 
for the exercising of the reasoning faculty, for the training of 
individual characteristics and differences. Being largely a 
fact subject, as now taught, it resolves itself mainly into a 
question of " yes " or " no," and this accentuates the prob- 
ability of failure. Individuals differ vastly in their ability to 
memorize, and therefore the poor memorizer is placed at a 
disadvantage. The pupil who can reason out a new demon- 
stration in geometry knows infinitely more geometry than 
he who can transcribe on paper every one of the prescribed 
demonstrations in a book on this subject. 

The Value and Place of Supervised Study. — This leads us 
logically to a discussion of the value of supervised study in 
mathematics. Unsupervised study is inefficient study be- 
cause much time and energy are lost in misdirected effort. 
Pupils do not know how to attack a lesson any more than they 
know how to perform the mechanical processes, until they are 
carefully taught. Class exercises avail little for the major- 
ity of the pupils because no two minds react in the same way. 
To clinch the class exercise individual guidance is required. 
The unsupervised recitation as a rule does not provide for 
this. Problems in algebra and originals in geometry are 
entirely dependent upon the characteristics of the individ- 
ual mind, which can be developed and trained only through 
the individual himself. To quote from an article by the 
author, 1 " the school must teach its pupils not to be perfect 

1 Journal of the New York State Teachers 1 Association, November, 19 18. 



6 Supervised Study in Mathematics and Science 

automatons, responding with machine-like accuracy to the 
whim of the examiner, but to become thinkers, with power 
and knowledge of how to attack and study out a problem, 
how to form personal opinions, how to get results, by them- 
selves. This, then, is the function of supervised study: to 
properly direct the pupil in his work so that he may develop 
the best methods of attacking problems ; that he may avoid 
wrong methods of reasoning; that he may most efficiently 
employ his time ; and that he may eventually acquire a power 
of skill that will classify him as a finished thinker, an educated 
man." 

Supervised study is only one of the several methods that 
need to be employed in bringing about a closer relationship 
between teacher and pupil and in the development of the 
pupil's native endowment in the field of mathematics. Such 
relationship might be illustrated as spokes of a wheel. Just 
as every spoke (Figure I) is necessary in the connection be- 
tween the rim which represents the pupil and the hub which 
represents the teacher, so supervised study should be given 
its proper position in the devices of the schoolroom. The 
other spokes, each with its peculiar evaluation, might be the 
recitation, the assignment, the equipment, tests and quizzes, 
standard tests and measurements, inspiration and sympathy. 

Division of the Course into Units of Instruction, Recitation, 
and Study. — In our discussion of the technic of the super- 
vised study of mathematics, let us first agree on our use of 
terms, as formulated by Professor Hall-Quest in his pioneer 
book, " Supervised Study " * and followed out in the other books 
in this series. In program of studies, let us include all the 
work offered in a school ; by curriculum, let us understand a 
1 Hall-Quest, "Supervised Study"; The Macmillan Company, 1916. 



Supervised Study Period in Mathematics 7 

group of subjects leading to a special end, as college prepara- 
tory curriculum, domestic science curriculum, etc. ; and by 
course, any single subject as algebra, civics, etc. Then, as a 
means of evaluating the course and giving it a definite and 




Figure I 



comprehensive development, we shall separate the subject 
matter into various units. By units of instruction, we shall 
mean the large topics around which the material revolves. 
In many cases these are the divisions noted in the table of 
contents ; they are the divisions of the subject into " type 



8 Supervised Study in Mathematics and Science 

lessons." 1 Such general topics as percentage, banking, 
factoring, graphs, would thus become units of instruction. 
Then the subdivisions of these larger units into smaller ones, 
around which one or more recitations would revolve, may be 
called units of recitation. These units may again be sub- 
divided into the work planned for a single day, or units of study. 

Types of Recitation. — In addition to this analysis of any 
course of study into its various units, the careful teacher will 
further decide on the technical form of presentation of each 
unit of study, or the type of lesson. Following the treatment 
of this phase as detailed by Strayer 2 and Earhart, 3 we may 
employ, as occasion prompts, the deductive, inductive, ha- 
bituation, expository, how to study, socialized, or review lessons. 
Since each type, however, has its peculiar aim and technic, 
the teacher will do well to make a careful study of them and 
of their application to the subject in hand. 

In general terms, the deductive lesson aims to draw forth 
new conceptions from our present knowledge. It is based on 
the process by which we think. The inductive lesson, on the 
other hand, leads up to new concepts by a series of successive 
steps, each definite and complete in itself. It is the process 
by which we accumulate knowledge. In the drill or habitua- 
tion lesson, the mechanical side of learning is stressed. By 
drill, needful automatic reactions are established. The ex- 
pository lesson seeks to make the new assignment as clear as 
demonstration and analysis make it possible. It is usually 
employed as a connecting link between the old and new mate- 

1 McMurry, "How to Study" ; Houghton Mifflin Co., 1909. 

2 Strayer, "Brief Course in the Teaching Process" ; The Macmillan Company, 
1912. 

3 Earhart, "Types of Teaching"; Houghton Mifflin Co., 1915. 



Supervised Study Period in Mathematics 9 

rial of a unit of study. The how-to-study lesson is self-explan- 
atory. Under review are usually included written or oral 
examinations. In a larger sense the review should cover the 
bringing together for periodic consideration, the clinching of a 
unit of recitation or unit of study. The socialized lesson may 
be poorly named from a standpoint of nomenclature ; possibly 
a better term at the present time would be the democratized 
lesson. At any rate, it is that type of lesson which introduces 
the human element into the school work. By it the work 
outside and that inside of the schoolroom are correlated. 
Although this type of lesson may be used sparingly, it is none 
the less important, and the teacher should feel that his greatest 
opportunity for reaching the consciousness of the child is 
presented through this form of lesson organization. Group 
assignments, dramatic productions, class programs, dual proj- 
ects, mathematical clubs, and like devices for impressing the in- 
terdependence of individuals in solving social or economic prob- 
lems, will tend to vitalize and democratize the subject matter. 
A more elaborate classification of exercise types as applied 
to the teaching of high school mathematics has been evolved 
in an illuminating article by Professor G. W. Myers, of the 
College of Education of the University of Chicago, in High 
School Mathematics and Science, June, 1921. In this article, 
the following types of mathematical class exercises are dis- 
cussed, and specifications or norms are given for judging each : 

I. The conceptual type. VII. The problem type. 

II. The expressional type. VIII. The topic type. 

III. The associational type. IX. The applicational type. 

IV. The assimilational type. X. The test type. 

V. The review type. XI. The research type. 

VI. The drill type. XII. The appreciational type. 



io Supervised Study in Mathematics and Science 

While space does not permit a detailed review of these, 
the article in question is commended to the reader for care- 
ful study. To treat it adequately here would be to quote it 
as a whole. 

The Time Schedule. — In order that the plan of supervised 
study may be carried out in its finest application, the period 
should be long enough for the pupil to do most if not all of 
his studying in school. This would mean a period of from 
ninety minutes to two hours in length and would also involve 
an extension of the school day in most cases. Superintendent 
L. M. Allen x remarks that a shorter time than the above is 
" neither hay nor grass," and that less than forty-five minutes 
for the study period itself will not suffice. The author of this 
book will grant that the longer the period the better the results, 
but from the experience in his own school for the past five 
years, he is constrained to disagree with the above conclusion 
of Superintendent Allen. In the school at Canton the periods 
are all one hour in length, the first thirty-five minutes being 
devoted to the recitation and the assignment, and the last 
twenty-five minutes being given up to the study of the lesson 
for the next day. Realizing that a longer study period would 
be very desirable, the author knows from experience that 
even this length of time will justify itself by increased and 
better work, as shown from the statistics as applied to the 
Canton school. 

It seems, therefore, only fair to conclude that, when it is 
impossible to increase the length of the periods to the limits 
suggested above, the installation of supervised study is still 
feasible and good results may be secured from the shorter 
period. In any case, the period will be divided into three 
School Review, June, 191 7. 



Supervised Study Period in Mathematics n 

parts: the review, the assignment, and the silent study. 
When the periods are sixty minutes in length, the approximate 
division of time among these three parts should be as follows : 

The Review 15 minutes 

The Assignment 20 minutes 

The Silent Study 25 minutes 

With a longer period, the study section will be increased 
more than the others. A sixty minute period means, of course, 
that only part of the study will be done in class ; a ninety 
or one hundred twenty minute period should make home 
work unnecessary. 

The Review. — The review will take the place of the old 
recitation and, while its length has been decreased, by inten- 
sive and well-applied questions the work ought to be thor- 
oughly covered in this time. The class review should es- 
sentially be a re-view of the difficult parts of the day's lesson ; 
and, while the weaker pupils will get the most attention, it 
will be a sort of summing up for all. It is unnecessary to re- 
view every minute step ; half of the usual recitation is spent in 
reciting on perfectly well-known and understood things. The 
review is not the time to show of what we know but to clear up the 
things we do not know or know only indistinctly. It should 
always be a real step forward, a sort of clearing house for the 
previous day's assignments. Again, the usual type of recitation 
is apt to be a kind of monologue with the teacher taking the 
leading part. As a matter of fact, the teacher should remain 
in the background. In war, the generals give orders but the 
rank and file does the fighting. The review, then, should be 
incisive, intensive, and conclusive. The entire class should 
not be held back by a few backward pupils. On the other 



12 Supervised Study in Mathematics and Science 

hand, the bright pupils should not be conciliated by insipid 
questions. The teacher should address the review to the 
weaker ones but by methods that will appeal to all. 

The Assignment. — The assignment is always the most 
important part of the recitation period. It should include, 
in addition to a definite allotment of new work, very clear 
explanations. The advance lesson should be carefully planned 
beforehand, so that there will be a definite amount of ground 
to be covered, a definite objective gained, and a definite advance 
made. If this is slurred over, the pupil will have no clear 
idea of what the lesson is about or of what he is to do. The 
assignment to be prepared should also be made with due 
consideration of its difficulty, the varying abilities of the 
members of the class, and the often overlooked fact that the 
pupils have lessons also in other departments. If all these 
elements have been taken into consideration and given thought- 
ful planning, the time allotted to this particular section of the 
period ought to be sufficient. But there must be plenty of 
time to cover the assignment fully and thoroughly; there- 
fore, it is better to assign too little than too much. In any 
quota of problems or examples, special difficulties likely to be 
encountered should be pointed out and possible methods of 
attack suggested. The assignment is perfect only when every 
pupil knows exactly what is the aim of the new work, what 
is the best method for its solution, and just what ground he 
is expected to cover before the next recitation. 

The Study Period. — The study section is the part devoted 
by the pupil to the study of the advance lesson, under the 
direct and sympathetic supervision of the teacher. In the 
sixty minute period used in the author's school, the pupils 
are not all expected to complete the assignment during the 



Supervised Study Period in Mathematics 13 

period. Some will, however, and it will be an incentive to 
all to strive to complete the work during the time allotted to 
study. At any rate, all will have been able to get a start and 
a start in the right direction. 

The teacher will have two classes of pupils to look after 
during the silent study period. One will be those who have 
some little technical difficulty with the new lesson. A well- 
directed question will usually set them on the right path. 
The other division will be those who are commonly considered 
failures, but who in many instances are simply pupils with 
some individual characteristics which react unfavorably for 
maximum efficiency. These pupils should be carefully studied 
by the teacher and their cases diagnosed. The problems 
thus presented to the teacher should awaken all of his deter- 
mination to solve them. The silent study period thus gives 
the teacher an opportunity for a study of the individual 
personnel of his class. The elimination of pupils from the 
list of failures should become the predominant effort rather than 
the elimination of pupils from class and school. Often a pupil, 
who would ordinarily have failed, has found himself through 
a little attention and study on the teacher's part during the 
study period. When the teacher can sit down with him, note 
his manner of work, detect his deficiences and weaknesses, 
and by a little tactful and sympathetic guidance, lead him 
into the paths of success, such a pupil will gain confidence and 
later economical independence. He must be taught how to 
walk — how to study. 

The Assignment Sheet. — The assignment sheet used in 
Canton, which is similar to the one described by Miss Simpson 
in her companion book in this series, 1 is reproduced at the end 

1 " Supervised Study in History" ; The Macmillan Company, 1918. 



14 Supervised Study in Mathematics and Science 

of this chapter. The object of the sheet is to induce the 
teacher to have a definite plan for each day's work. Work not 
carefully planned is apt to be poorly done. Nothing begets 
carelessness and indifference on the part of the class so much 
as a lack of purpose and plan on the part of the teacher. 
These sheets need take scarcely any time ; in fact, they will 
save time, because the teacher will know exactly what he is 
going to do, what material he is going to use, and where it is 
to be found. In addition thereto, he will have at the close 
of the term a complete record of work accomplished. 

How to Make a Lesson Plan Sheet. — Under Review note 
exactly the things that need to be reemphasized. The ref- 
erences to other books for supplementary material may be 
noted under Memoranda. A good scheme is to write on the 
back of the assignment sheet the names of the pupils who 
should receive especial attention during this review. 

The Threefold Assignment. — As Professor Hall-Quest 
explains in his book, the assignment should be in three parts : 
one to take care of the inferior pupil, one to take care of the 
average pupil, and one to take care of the superior pupil. 
Hence we have the three part assignments, or the minimum- 
average-maximum plan. 

The minimum assignment should cover the minimum es- 
sentials, i.e. the work that all must do at the very least, and 
that the majority can easily do within the twenty-five minutes 
allotted to study. It should be so planned that pupils who 
never do any more than this amount will be able to pass the 
final examination, which is or should be the minimum require- 
ment, but too often becomes the only aim of the teacher. 
These pupils will not obtain a high mark, but they will have 
mastered enough of the subject to get a passing grade. The 



Supervised Study Period in Mathematics 15 

object of the average and maximum assignments is to produce 
pupils who will not only pass but pass high. 

The average assignment should include more examples and 
different kinds of examples but not necessarily of a much 
harder nature. However, the more problems a pupil can 
solve correctly in a given time, the more skilled he will be- 
come, up to a certain limit. 

The maximum assignment is to take care of the brighter 
pupils, — those who are able to do more than the average 
pupil and who should have some incentive to do advanced 
work. Usually this assignment will be given from other texts 
and will consist of more difficult material. This section 
should be so limited in amount as not to discourage but to 
incite to a desire to cover it. It will incidentally keep a 
disturbing pupil out of mischief. But the tasks should be 
constructive and not simply the old fashioned " busy " work 
which employs but does not develop. 

Under Study note any points to be kept in mind while the 
pupils are working, especially in regard to mistakes they are 
likely to make. This section of the sheet may be made very 
effective, if the teacher is fully alive to the function of the 
supervised study idea. In the course of a year this plan will 
have become a real series of methods. 

The loose-leaf notebook makes an excellent method of 
preserving the sheets. They will be found of inestimable 
value another semester. 

How to Use the Assignment and Study Sheet. — Do not 
become a slave to the sheet; make it your servant. If it 
means omitting your daily recreation, eliminate it, not your 
recreation. It ought not to take much time and, as has been 
suggested, it will eventually be found to save time. 



1 6 Supervised Study in Mathematics and Science 

Strive always to inspire the class to reach the maximum 
assignment, to raise the maximum and lower the minimum end. 

Determine the proper proportions of pupils who should get 
the various sections. The following is a normal distribution : 
80 per cent should complete the average assignment, 10 per 
cent the minimum only, and 10 per cent the maximum. 
When these percentages vary greatly from the established 
norm, the nature and length of the assignments should be 
modified proportionately. That is, in a class of 30 pupils, 24 
should do all the average assignment before the end of the 
period. Three will be behind and will need special attention 
next day ; three will be working some of the examples in the 
maximum assignment. This of course is subject to varia- 
tion from day to day, but may serve as a guide. 

At the close of the period, the teacher should ascertain how 
many of the class have completed the minimum, the average, 
and the maximum assignments. This may be done by having 
all hand in their papers a few seconds before the period 
ends. The pupil may note on his paper which assignments 
he has completed. What remains of the assignment may be 
required of the pupils the next day. Various schemes may 
be evolved ; some will be touched upon in connection with the 
illustrative lessons in Part One. 

The assignment should be written each day upon the board 
and numbered I, II, III to correspond with the different 
assignment groups. The pupils need not be told the reason 
for this as it will be better if they do not understand the 
reason for the differentiation. The pupils should form the 
habit of noting down the assignment, thus getting practice 
in keeping a written record of important facts and engage- 
ments. 



Supervised Study Period in Mathematics 17 

The Management of the Supervised Study Period. — As 
soon as the class meets, take the roll call and at once start 
the review. The time spent in taking the roll should be 
reduced to a minimum ; certainly not over two minutes at the 
most. An excellent method is to have on a sheet of paper a 
diagram representing the seats and in each space, which 
represents a seat, put the name of the pupil who occupies it 
each day. Then those absent can be quickly noted through 
the vacant spaces, and their names checked. 

The time allotted to the review and the assignment during 
the first thirty-five minutes should be held to as closely as 
possible ; a schedule or time table is absurd unless adhered to 
conscientiously. At the end of the thirty-five minutes, a 
bell may be rung simultaneously in all rooms from the study 
hall or some other central place, and the next two minutes 
devoted to physical drills, setting-up exercises, etc. 

Then, for the next twenty-five minutes or fraction thereof, 
the pupils should be required to work on their new lesson. 
The teacher should insist that no other work be done during 
the study period, until the lesson in hand is finished. At 
first, some may try to study the succeeding lesson but a little 
firmness will soon bring desired results. 

As soon as possible after the class has been organized, it 
may be expedient to reseat the pupils according to their abili- 
ties ; those of minimum ability on one side of the room, those 
of maximum ability on the opposite side, and the average 
pupils between. This allows the teacher to come into closer 
contact with the weaker pupils, and he can give them addi- 
tional attention. This classification must be done with tact 
so as not to hurt any child's feelings ; therefore, the earlier in 
the term it is done the better. It is not necessary to name 



1 8 Supervised Study in Mathematics and Science 

the groups. Calling them A, B,C without any further char- 
acterization will suffice. 

Some method of checking the results of the work accom- 
plished during the period should be worked out. Various 
schemes will be mentioned in connection with the illustrative 
lessons ; but a few general remarks may be made here. Any 
plan which would depend entirely on the amount of work done 
in class must take into consideration the fact that some 
children are accurate but slow ; these must not be discouraged 
by low marks. While rapidity is desired, accuracy is more 
important, and these pupils should be encouraged to make an 
effort to maintain accuracy and secure greater rapidity. On 
the other hand, it is very important that as much of the work 
be done in class as practicable so that the teacher may know 
that the work is the pupil's own. Thus there is presented to 
the teacher a fine question for analysis : to discover why a 
pupil does not accomplish so much as is expected. The teacher 
should be untiring in his effort to solve such a problem. 
Theoretically the pupils who habitually solve only the first 
or minimum assignment should receive a mere passing grade, 
those doing the average assignment should receive marks 
between 80 and 90, and those doing the maximum amount 
should receive honors. But this must eventually be deter- 
mined by the ability of the pupil to work similar exercises in 
the review, by his lack of dependence upon the teacher during 
the study period, and to a lesser extent by the periodic test. 



Supervised Study Period in Mathematics 

ASSIGNMENT AND STUDY SHEET 
Subject..... Period .Date 

Unit of Instruction 

Unit of Recitation.. 

Unit of Study 

Lesson Type 



19 



Review: 


Memoranda 


Assignment : 




1. Minimum 




2. Average 




3. Maximum 




Study: 





Number of pupils solving minimum assignment... 

Number of pupils solving average assignment 

Number of pupils solving maximum assignment. 

Total . 
Figure II 



FIRST SECTION. ALGEBRA 

CHAPTER TWO 
DIVISIONS OF ELEMENTARY ALGEBRA 

It is advisable in any course of study that the teacher have 
a definite outline of the successive stages in the development 
of the subject, and that these be further subdivided into their 
smaller units. These general topics may be called Units of 
Instruction and the subtopics, Units of Recitation. The 
course of study here outlined is that suggested by the New 
York State Education Department in its 1910 syllabus. 

A. Units of Instruction. 

I. Introduction. 

II. Positive and negative numbers. 

III. Addition. 

IV. Subtraction. 
V. Multiplication. 

VI. Division. 

VII. Factoring. 

VIII. Common factors and multiples. 

IX. Fractions. 

X. Simple equations. 

XI. Graphic representation. 

XII. Involution. 

XIII. Evolution. 

XIV. Radicals. 

XV. Quadratic equations. 

B. Units of Recitation. 

20 



Divisions of Elementary Algebra 21 

The subdivisions do not imply that only one recitation is to be 
given to each topic, but rather that all the recitations for the length 
of time needed will center around this special topic. 

I. Introduction. The material given under this head 
varies in different texts, but it will at least contain : 

Units of Recitation : 

1. Symbols of algebra. 

2. Literal numbers. 

3. Historical notes. 

4. Definitions and notation. 

II. Positive and Negative Numbers. 
Units of Recitation: 

1. Explanation and illustration of signed numbers. 

2. The addition, subtraction, multiplication, and 

division of signed numbers. 

III. Addition. 

Units of Recitation: 

1. Addition of monomials. 

2. Addition of polynomials. 

IV. Subtraction. 
Units of Recitation : 

1. Subtraction of monomials. 

2. Subtraction of polynomials. 

3. The parenthesis. 

V. Multiplication. 

Units of Recitation : 

1. Multiplication of monomials by monomials. 

2. Multiplication of polynomials by monomials. 

3. Multiplication of polynomials by polynomials. 

4. Special cases. 



22 Supervised Study in Mathematics and Science 

VI. Division. 

Units of Recitation: 

i. Division of monomials by monomials. 

2. Division of polynomials by monomials. 

3. Division of polynomials by polynomials. 

VII. Factoring. 

Units of Recitation: 

1. To factor a monomial. 

2. To factor a polynomial. 

3. To factor a polynomial whose terms may be 

grouped to show a common polynomial factor. 

4. To factor a trinomial which is a perfect square. 

5. To factor the difference of two squares. 

6. To factor a trinomial in the form of x 2 -\-px+q. 

7. To factor a trinomial in the form of ax 2 +bx+c. 

8. To factor the sum or difference of cubes. 

9. To factor the sum or difference of the same odd 

powers of two numbers. 

10. To factor the difference of the same even powers 

of two numbers. 

11. To factor by the factor theorem. 

12. To factor by special devices. 

VIII. Common Factors and Multiples. 
Units of Recitation: 

1. Highest common factor. 

2. Least common multiple. 

IX. Fractions. 

Units of Recitation: 

1. Reduction to higher or lower terms. 

2. Reduction to an integral or mixed expression. 

3. Reduction to similar fractions. 



Divisions of Elementary Algebra 23 

4. Addition of fractions. 

5. Subtraction of fractions. 

6. Multiplication of fractions. 

7. Division of fractions. 

8. Complex fractions. 

X. Simple Equations. 

Units of Recitation : 

1. Equations with one unknown. 

2. Equations involving fractions. 

3. Literal equations. 

4. Problems. 

5. Simultaneous equations. 

a. Elimination by addition or subtraction. 

b. Elimination by comparison. 

c. Elimination by substitution. 

6. Literal simultaneous equations. 

7. Problems. 

8. Equations with three or more unknowns. 

9. Problems. 

XI. Graphic Representation. 

Units of Recitation : 

1. Graphs of statistics. 

2. Graphic representation of linear equations. 

XII. Involution. 

Units of Recitation : 

1. Involution of monomials. 

2. Involution of polynomials. 

3. Involution by the binomial theorem. 

XIII. Evolution. 

Units of Recitation : 

1. Evolution of monomials. 

2. To extract the square root of polynomials. 



24 Supervised Study in Mathematics and Science 

XIV. Radicals. 

Units of Recitation: 

i. To reduce radicals to their simplest form. 

2. To reduce a mixed surd to an entire surd. 

3. To reduce radicals to the same order. 

4. Addition and subtraction of radicals. 

5. Multiplication of radicals. 

6. Division of radicals. 

7. Involution and evolution of radicals. 

8. Rationalization. 

9. Square root of a binomial quadratic surd. 
10. Radical equations. 

XV. Quadratics. 

Units of Recitation: 

1. Pure quadratic equations. 

2. Affected quadratic equations. 

3. Solution by 

a. Factoring. 

b. Completing the square. 

c. The formula. 

4. Literal quadratic equations. 

5. Radical equations in quadratics. 

6. Problems. 

7. Simultaneous equations involving quadratics. 

a. One simple equation, one involving the second 

degree. 

b. Two homogeneous equations of the second 

degree. 

c. Symmetric equations of third or fourth degree, 

readily solvable by dividing the variable 
member of one by the variable member of 
the other. 

8. Problems. 



Divisions of Elementary Algebra 25 

Time Table for the Term. — Below is a suggested time 
table for the year's work : 

ALGEBRA : 40 weeks 
First Term 

1st week Introduction 

2d week Addition 

3d, 4th, 5th weeks Subtraction and parenthesis 

6th and 7th weeks Multiplication 

8th and 9th weeks Division and review 

10th to 15th week Factoring 

1 6th week Common factors and common multiples 

17 th to 20th week Fractions 

Second Term 

21st to 24th week Simple equations 

25th week Graphs 

26th and 27th weeks Involution 

28th and 29th weeks Evolution 

30th to 34th week Radicals 

35th and 36th weeks Quadratics 

37 th to 40 th week Review 

Factors Modifying the Foregoing Arrangements. — Local 
conditions will necessarily determine the emphasis to be placed 
on the respective units of subject matter. If a course in 
general mathematics is followed, it is obvious that considerable 
modifications will be required. Some classes are more mathe- 
matically-minded than others and this fact should affect 
emphasis. Whatever units are found to be rarely of value 
even in advanced studies of mathematics should be practically 
ignored. There is not time in public school work for the 
elaboration of the useless. Many textbooks in mathematics 
are now so arranged that certain designated sections may be 
omitted without affecting the continuity. 



26 Supervised Study in Mathematics and Science 

LESSON I 

THE INSPIRATIONAL PREVIEW 

Purpose. — The purpose of such a lesson is to arouse in the 
child the will to learn, to awaken interest in the study of 
algebra, and to outline in a general way some of the things of 
interest which will be studied during the course. 

Need. — Coming from the grades with no very clear idea of 
what high school means and assigned to new subjects, the 
very names of which are often strange, it is no wonder that the 
young pupil is not only lacking in any particular interest in 
his new work, but may even have a natural antipathy toward 
it from the start, unless interest be aroused through this 
inspirational preview. 

It is important in meeting any class for the first time that 
the teacher get en rapport with the pupil as quickly as possible. 
It is well in place to give a simple talk on the history, practical 
value, and general content of the subject, — a sort of advertis- 
ing or " selling" talk. The careful traveler will always plan 
his trip ahead in order that he may be prepared to note all the 
important and interesting things that may lie in store for him ; 
otherwise, many things would escape his attention. So the 
preview is a sort of bird's-eye view of the course, a cranking 
up of the pupil's interest, preparatory to a good start and a run. 

Method. — Simple language should be used. The class is 
composed of immature boys and girls to whom big words and 
phrases mean little ; the talk should be more to the child 
than about the subject. The essential thing is to make the 
children feel at home, to arouse enthusiasm for the subject, 
and to make them look forward to their work in algebra with 
pleasure. 



Divisions of Elementary Algebra 27 

When a strange word or important date is given, it will have 
more effect upon the class if it is written upon the blackboard 
at the time that it is mentioned. Prearranged work upon 
the board, simply referred to in passing, does not rivet their 
attention so well. 

When the preview is completed, it might be well to ask a few 
questions concerning what has been said and give, if neces- 
sary, any further details needed to make the subject clear. 
Ask the pupils to jot down the data you have placed upon 
the board and tell them to hand in the next day a simple state- 
ment of what has been said. This is not so important from 
the standpoint of the material as it is from the standpoint of 
inculcating at once the necessity of paying attention, of being 
specific regarding facts ; and of the implication that mathe- 
matics is closely correlated with English in the clearness of 
its exposition. 

Historical. — A little of the history of the subject will 
arouse the pupils' interest in the age and romance of algebra. 
It is best not to go much into detail because of the complexity 
of its historical development. It will be better also to intro- 
duce each new topic, when studied, by its individual history. 
Many of the recent texts in algebra have historical notes 
scattered through the book as a help to humanizing the sub- 
ject. Pictures of famous mathematicians add to the interest 
of these historical references. A framed portrait of one or 
two mathematicians of note, or a statuette, placed in the room, 
will give an added atmosphere to the study of the subject. 

Origin of the Word "'Algebra" Its name is derived from the 
title of a book which the Arabs introduced into Europe in the 
ninth century. The full title of the book was " Al-jebr w' al 
muqabalah," of which the first two syllables have been cor- 



28 Supervised Study in Mathematics and Science 

rupted into the present term, algebra. In the original tongue 
it referred to the process of transposing terms in the manipu- 
lation. Hence the solving of problems was early considered 
the main business of this subject. 

Contributors to the Subject. Modern algebra has not come 
down to us in its present composition, but like the automobile 
and every other invention, it is the result of years of growth 
and of contributions of many minds. The early Egyptians 
and the ancient Greeks of the " golden age" had some con- 
ception of the equation and have left their imprint upon its 
development. Heron of Alexandria about ioo B.C. made the 
greatest advance in its development up to that time, but 
Diophantus, a fourth-century Greek, was the first to write an 
entire book upon the subject. He emphasized indeterminates, 
which are even now called Diophantine after him. But as 
stated above, the book whose title has given the subject its 
modern name was the first general treatise of importance. 
The modern founder of algebra was a Frenchman, named 
Vieta, who, in 1591, gave the science the technical symbolism 
which is in use to-day. Among other modern contributors 
were Wessel of Norway, Gauss of Denmark, and Sir Isaac 
Newton of England. 

Interesting Incidents. The history of the subject abounds 
in many interesting episodes. It is said 1 that Sir William 
Hamilton, an Englishman, who had been working for years on 
a certain problem, was one day taking a stroll with his wife, 
when the solution suddenly flashed into his mind, and he at 
once engraved upon a stone in Brougham Bridge, which he 
was crossing at the time, one of the fundamental formulas of 

1 " Science-History of the Universe," VIII; M athematics, Current Litera- 
ture Publishing Company, 191 2. 



Divisions of Elementary Algebra 29 

modern algebra, called quaternions. This bridge has ever 
since been called Quaternion Bridge. 

An Italian named Tartaglia, who claimed a certain alge- 
braic discovery, was challenged by another famous mathe- 
matician by the name of Fiori. The contest was to see which 
one could solve the greatest number of a collection of thirty 
problems within thirty days. Tartaglia, by using his new 
discovery, which by the way is now a matter of common 
knowledge, i.e. the cubic equation, solved the entire thirty 
within two hours' time. He celebrated his triumph by com- 
posing some verses, but, according to the custom of the times, 
he kept his discovery a secret for many years. 

Practical Value of Mathematics in General and Algebra in 
Particular. — From the very earliest times, the study of 
mathematics has been considered primarily practical. Mathe- 
matics has been, in some form or other, the bulwark of the 
education of the Chinese, the Arabian, the Assyrian, the Jew, 
the Greek, the Roman, and every modern race. One might 
as well try to conceive of life without atmosphere as to try to 
separate the influence of mathematics from human life and 
endeavor. 

Dr. Eugene Smith of Columbia University says that " if 
all mathematical knowledge were eliminated, civilization 
would be demoralized, factories would stop for want of ma- 
chinery, and life would revolt to chaos." * 

This is the era of machinery. A machine, mathematically 
wrong, is a failure. Long before the machine is put on the 
market, however, it has been the subject of painstaking effort 
on the part of the inventor, the draftsman, the pattern- 

1 "Mathematics m Training for Citizenship"; Teachers College Record, 
May, 191 7. 



30 Supervised Study in Mathematics and Science 

maker, the mechanic, and the promoter. Each one has applied 
his knowledge to his labor, and the finished product is the re- 
sult of the accumulated researches and experiences of these 
men in turn. Imagine the invention of the steam engine, 
the sewing machine, the typewriter, the adding machine, the 
lathe, the printing press, the automobile, the airplane, with- 
out an expert mathematical training on the part of the 
inventors. 

The recent war with its wonderful though terrible inventions, 
which sprang into being from all sides, illustrates most forcibly 
the value of mathematical ability, because every machine from 
the gas mask to the submarine involved the exercise of mathe- 
matical genius. From the moment of its first conception in 
the mind of the inventor to the actual firing of the first shot 
on the battlefield, the giant field gun is a product of mathe- 
matics. The battle of Messines Ridge was won by engineers 
who skillfully tunneled the hills. Dr. Nichols, of the Uni- 
versity of Virginia, contributed to our wonderful shipbuilding 
exploit by successfully solving an equation to the ninth degree. 1 

Moritz shows our commercial dependence on mathematics. 
Thales, the ancient mathematician, was by profession a mer- 
chant, and yet he studied mathematics for the intrinsic value 
it held for him. Such modern business problems as equation 
of payments, theory of interest, valuation of debenture bonds, 
amortization of interest-bearing notes, life insurance mor- 
tality tables, distribution of dividends, casualty insurance, 
and a thousand others, have their solution through the ap- 
plication of pure mathematics. Statistical work, which is 
used in kundreds of transactions and enterprises of every 
kind, is all mathematics. Business executives say they prefer 
1 C. E. White in School Science and Mathematics, January, 1919. 



Divisions of Elementary Algebra 31 

mathematically trained men because they are more methodical, 
exact, and resourceful, and, therefore, efficient. 1 

Algebra Has an Important Position. From a strictly 
utilitarian standpoint, algebra has a firm claim for an im- 
portant position in our program of studies. It offers the only 
means of solving many of the problems connected with 
engineering, architecture, navigation, surveying, meteorology, 
geology, physiology, and even psychology, if we accept the 
Weber-Fechner law, 2 astronomy, physics, chemistry, and 
many other branches. No mechanic or artisan can read 
intelligently his trade journal, a technical book, an article 
in the encyclopedia, without a knowledge of the universal 
language of algebra. There are 27,000 volumes in the Naval 
Observatory at Washington which absolutely require a 
mathematical training for their perusal. 3 Statistics of all 
kinds are given in equations or mathematical formulas. The 
graphical treatment is employed by the economist, the business 
expert, the physician, the dietitian, and men of hundreds of 
other professions. The handling of pig iron in the modern 
foundry is a result of long continued analytical experiments 
based on algebraic formulas. 

Algebra Is Necessary to Secure a Higher Education. It is 
indispensable to the future student of astronomy, physics, 
chemistry, and higher mathematics. The modern engineering 
world and all technical schools are forever closed to him with- 
out a mastery of this subject. Just the other day the writer 
had an illustration of this. A young man who had been 

1 R. E. Moritz, "Our Relation of Mathematics to Commerce"; School 
Science and Mathematics, April, 19 19. 

2 C. E. White in School Science and Mathematics, January, 1919. 
N 3 Schultze, " The Teaching of Mathematics " ; The Macmillan Company, 19 12. 



32 Supervised Study in Mathematics and Science 

recently graduated from high school found that it was possible 
for him to go to college but he had been allowed to go through 
this school without algebra and geometry, and he found to his 
dismay that he was unable to matriculate because of this 
deficiency in his education. Many a young man seems for- 
ever doomed to a nominal wage and a position near the foot 
of the ladder because he failed to fit himself for higher positions 
by the careful study of mathematics. 

A noted inventor once said that he had no use for algebra 01 
higher mathematics, but in the next breath he admitted that 
he hired experts to work out for him all problems involving 
advanced mathematics. Mathematical physics and mathe- 
matical chemistry are important branches of science to-day, 
and every phase of electrical engineering is intimately bound 
up with algebra and its manipulations. 

The writer once knew a man who lost a fine position as 
assistant contractor because he did not know algebra, al- 
though he was an expert workman. His would-be employer 
felt he could not afford to have a man in this responsible 
position who was ignorant of this subject, as occasions some- 
times arose which demanded a knowledge of it. 

A man must know algebra to advance in the automobile 
business. A modern blue print with its mechanical and 
mathematical symbols looks like a Chinese puzzle to the lay- 
man, but to the trained mind of the skilled mechanic it is as 
plain as the printed page. 

And so we might multiply the examples of the practical 
value of algebra indefinitely, but enough has been said to 
arouse in the mind of the pupil the strong suggestion that here 
is a subject which has a bearing, both directly and indirectly, 
upon his future as well as upon the progress of the world. 



Divisions of Elementary Algebra 33 

Bird's-eye View of the Course. — A few minutes might 
profitably be spent in outlining the semester's work. Tell 
the class that there will be a certain amount of formal work, 
similar to that done in arithmetic, such as learning how to 
add, subtract, multiply, and divide algebraic terms and quan- 
tities, finding factors, reducing fractions, and manipulating 
equations, in order that we may arrive at the solution of 
problems. Explain that this is necessary in order to become 
familiar with the tools of algebra, just as the carpenter must 
learn how to use the hammer, the saw, the square, and the 
compass, before he can build a house. 

Explain that from time to time, as a topic is finished, we 
shall have a sort of field day, when we may exhibit our work, 
give special reports of certain phases of the subject, and 
invite our friends to see the things we have accomplished. 

Tell the pupils also that we shall occasionally have contests, 
in which we shall choose sides to see which side can win. This 
may be done with fine results, as will be illustrated later, upon 
the completion of the work in factoring. 

Again, explain that during the year the class will learn how 
to solve various problems which will be drawn from business, 
agriculture, the vocations, etc. Tell them that they them- 
selves will also be expected to make up and solve problems, 
and that data drawn from current events, like an election or 
a ball game, will be used. 

Conditions for a Successful Preview. — Too much must not 
be attempted in this first lesson. The word " inspirational " 
very graphically illustrates the underlying motive for this 
lesson. The preceding material may well be added to or in 
part eliminated, as seems best in the judgment of the teacher 
himself. It is given simply to suggest some of the things that 



34 Supervised Study in Mathematics and Science 

may make this first meeting of the class an inspiration and a 
forward look. 

Illustrations concerning the practical value will have more 
force if they have come under the actual observation of the 
teacher. Indeed, the teacher of algebra may well keep a note- 
book in which such material and experiences may be collected 
and added to from time to time. Such a teacher will soon 
accumulate a valuable set of illustrations, and one that will 
have variety and modern application. The historical notes will 
also vary with the teacher ; he must ever be on the alert for 
interesting incidents to use in this connection. Such material, 
aside from the cases which come under his own observation, 
will be found in magazine articles, newspaper articles, and 
the experiences of others and will be drawn from the pupils 
themselves. 

LESSON II 

UNIT OF INSTRUCTION I. — INTRODUCTION 

Lesson Type. — A Lesson in Correlation 

Program or Time Schedule 

The Review 5 minutes 

The Assignment 30 minutes 

The Study of the Assignment 25 minutes 

Purpose. — The purpose of this lesson is threefold : (1) to 
gain the confidence and free expression of the pupil by getting 
him to do something he knows how to do ; (2) to review and 
reemphasize some of the fundamental operations ; and (3) to 
link together arithmetic and algebra by showing their inter- 
relationship. 



Divisions of Elementary Algebra 35 

The Review. — Since this is the first review work of the 
year and there has been no assignment, a few questions like 
the following might be asked, suggested by the " Inspirational 
Preview " : 

How did we get the word algebra t 

Name some countries which have contributed to the devel- 
opment of this subject. 

For what is Vieta important ? 

Relate an interesting incident connected with the history of 
this subject. 

Name some professions which presuppose a knowledge of algebra. 

The Assignment. 

1. Information given by the teacher regarding the (a) func- 
tion and (b) applications of algebra. 

2. Review of the fundamental processes in arithmetic. 

3. Recognition of the interdependence of algebra and 
arithmetic. 

The Function of Algebra. Sir Isaac Newton called algebra 
" Universal Arithmetic." Comte defined arithmetic as the sci- 
ence of values and algebra as the science of functions. Alge- 
bra deals historically and primarily with number. Primeval 
man, desiring to count his possessions, used various forms of 
tallies. The ancient Roman used the pebble, setting aside one 
for each article counted. The ten fingers or digits were early 
used in counting, thus giving us the term used in numeration, 
and later evolving into the ten digit or decimal {i.e. decern, ten) 
system. Gradually numbers became of interest because they 
allowed combinations. We therefore use symbols to stand for 
or represent things, later substituting the thing itself. Thus 
we are enabled to derive a general statement which may be 
applied to all similar cases. 



36 Supervised Study in Mathematics and Science 

We have done something of this in our arithmetic when 
we used the statement BxR equals P. Any problem in 
percentage may be reduced to this formula or some variant of 
it. When the substitutions are made, the problem may be 
solved. As in the case of percentage we have used the first 
letter of the word indicated, so in algebra we also represent the 
unknown by letters. But in that case, not knowing what the 
words may be, we do not try to use the initial letter. It has 
become customary to employ the least used or last letters of 
the alphabet, x, y, and 2. 

We find, therefore, that algebra becomes a general science 
while arithmetic remains a particular science, and, though 
they may be said to resemble each other in some respects, in 
reality algebra becomes a new and more general method of 
manipulating quantities. 

Applications of Algebra. The fundamental processes of 
addition, subtraction, multiplication, and division of whole 
numbers and fractions form the foundation stone of the 
formal work in algebra as well as in arithmetic, but we shall 
find that their functions and applications in this new science 
reach a breadth which is impossible in arithmetic. In solv- 
ing problems in arithmetic we are always limited to things 
concrete, but in algebra, through the use of the unknown, we are 
at liberty to sweep the whole field in our solution. It is for this 
reason that algebra assumes a universal value of its own. 

Review of the Fundamental Processes in Arithmetic. — A 
half hour's time may well be spent in reviewing the funda- 
mental operations in arithmetic and in bringing out the errors 
which are very common to most first year pupils, such as the 
product when multiplying a number by unity or zero, the 
manipulation of simple fractions, mixed numbers, reduction 



Divisions of Elementary Algebra 37 

of a fraction to unity in cases like i = i, cancellation, etc. It 
will amaze one to find out the number of pupils who, sup- 
posedly ready for algebra, cannot do correctly some of the 
simplest fundamental operations in arithmetic. It will sur- 
prise not only the teacher but the pupil as well, since he is 
apt to feel that, having passed the final preacademic examina- 
tion in this subject, he is somehow endowed with a supreme 
and unfailing knowledge of arithmetic for all time. 

The teacher may apply some of the operations of arithmetic 
to algebra, but it is best not to do too much of this at first. 
For instance, after the fraction -J has been reduced to 1, we 
might take -^ and show that it reduces to ix, or after a review 
of the multiplication of fractions, as JXi = i, we might show 
thatiXf = f, keeping in mind that we are not at present 
interested so much in the teaching of algebra as we are in 
showing that our present knowledge of arithmetic will be of 
constant help to us ; and in emphasizing that we are not going 
to work in a strange land but will have with us old friends. 

Interdependence of Algebra and Arithmetic. The nomen- 
clature is similar to that of arithmetic and even the mechani- 
cal method of setting down the problem, as in division, will 
be the same. 

We have already used the simple equation in the lower 
grades when in teaching addition we give such problems as 
" two and what make five? " In arithmetic we call this the 
Austrian method, but in algebra it becomes finding the un- 
known. 

We have learned that we cannot add together 5 apples and 
3 oranges except to say that we have 5 apples plus 3 oranges. 
So in algebra we cannot add $a and 36 except by indicating 
$a plus 36. 



38 Supervised Study in Mathematics and Science 

There is an intimate relation between exercises in removing 
the parentheses in arithmetic and algebra, such as (18 — 2 
4-4X2), in which the order of performing the operations and 
rules for removing the signs of aggregation are identical. 

In the use of the question mark to indicate what is desired, 
we have in arithmetic simply anticipated the use of the un- 
known x in algebra, in such an example as this : 

3*5 = 1 
8 8 8 

The graph is used in many exercises in arithmetic to show 
data which are used in the problem. These statistical graphs 
are a very important phase of elementary algebra. (See 
Unit of Instruction XI, Chapter Two.) 

We have already referred to the use of the formula in 
arithmetic. In addition to BxR equals P, we have a number 
of others relating to problems in interest, as P equals 
I-7-($iXrXt). Again, in measurement we have A equals 
bXh] circumference of a circle, or 0, equals irXD. 

Many texts in arithmetic also to-day have a section covering 
simple linear equations of one unknown which are solved 
algebraically. 

The Study of the Assignment. — I or Minimum Assignment. 

Ten or more examples to illustrate the common errors made 
in arithmetic as suggested in the paragraph on fundamental 
processes. 

1. 48X0= 7. i+i+f = 

2. oXio|= 8. 4i-2^ = 

3. 4|X5t^= 9. 12-4X3+6 = 

4. f+|+2i= 10. *S+3i = 

5. 1*1= 11. 

6. fXiX| = 



Divisions of Elementary Algebra 39 

II or Average Assignment. 

Eight or ten examples to illustrate the similarity of arith- 
metic and algebra as explained in the paragraph on inter- 
dependence. 

12. 32 and what make 50? 

13. 17 and (%) make 25, what is x? 

14. 23 and 32 give? 

15. 67 times 7 give (#)? 

16. 7 books plus 3 chairs plus 2 books plus 6 chairs equal (?) 

books plus (?) chairs. 

17. 5 b+ 3 c + 2b + ic = (?) b+(?)c. 

18. 2> x + 7 X = h° w man Y x ? 

19. If & = 5 and area = 120, find the altitude in formula, area = 

bXh. 

III or Maximum Assignment. 

20. Mention some other similar cases of algebraic applications 
in arithmetic. 

21. Give some formulas besides those stated, which you have 
had in arithmetic. 

22. Bring to class some illustration of the use of the graphic 
means of showing statistics. 

23. Report on the origin of symbols of operation. (Slaught 
and Lennes' " Elementary Algebra," p. 7, and Hawkes, Luby, 
and Touton's " First Course in Algebra/' pp. 4, 5.) 

The Silent Study Period. — As soon as the time for the 
study period arrives, the pupils should commence work upon 
the lesson assigned for the next day. This assignment in 
three sections has been fully explained in Chapter One. The 
complete assignment should always be placed upon the board 
before the class assembles, so that there may be no delay in 
commencing to work. 



40 Supervised Study in Mathematics and Science 

Since the exercises above suggested are simple, the pupils 
will probably not have much trouble, but they should under- 
stand that, in case anyone finds difficulty with the work, he 
has the privilege and is expected to raise his hand for aid. 
The teacher may then step quietly to his desk and find out 
what the difficulty is. 

It is important that the teacher and the pupil realize that 
the teacher is not expected to do the pupil's work for him. 
The teacher's part is to direct attention to his difficulty. The 
obstacle should be skillfully cleared up through the redirected 
effort of the pupil along the right path. 

If the point raised seems likely to be a stumblingblock to 
others, the teacher may step to the board and, calling the 
attention of the class to the difficulty, make necessary expla- 
nations which will at once be a help to all. It often happens 
that some little point was overlooked in the explanation before 
the class, which may now be made clear. 

Encourage the pupils to do as much of the assigned lesson 
as possible during the study period. The more the pupil does 
in the classroom, the better will he understand his work. 

Summary on the Review. ■ — Each day should provide some 
definite review of the preceding day's work, some definite 
advance in the mastery of the subject, and some definite work 
assigned. The pupil then becomes conscious of something 
positive having been accomplished. Nothing is more detri- 
mental to the morale of the pupil than for him to feel that a 
day's work or a recitation period has been wasted or, at 
least, has passed without some definite advance. When our 
pupils realize that nothing will be allowed to interfere with the 
day's work, we shall find that they will be more anxious to 
eliminate their absences. Friday or the day before or after 



Divisions of Elementary Algebra 41 

a vacation or some circumstance is allowed often to break 
up the routine of steady, purposeful work. With the proper 
attitude and evaluation of each day's importance, however, the 
teacher can make every meeting of the class, no matter under 
what disadvantages, a distinct step forward. The morale 
of the allied army was at high pitch until the Rhine was 
reached, and then it became a matter of anxiety to the com- 
manders, because the soldiers felt that their object had been 
attained and that, with no further advance being made, they 
were merely marking time. 

The review should be short, snappy, and purposive. It 
really takes the place of the old recitation, as such, and since 
it is much shorter in time, there must be intensive work done. 
It should be a re-view of the previous day's work, a clearing 
house for all the difficulties encountered in the study of the 
assignment, and an opportunity for the pupils to view from a 
new and broader angle the work studied the preceding day. 

Summary on the Assignment. — This is the portion of the 
period devoted to the explanation of the new lesson. The 
teacher should do most of the board work, the pupils following 
the operations at their seats. As far as possible the work 
should be developed through the pupils, since they will th«n 
become active participants and their interest peculiarly acute. 
Except on special occasions, all board work should be elimi- 
nated, so far as the pupils are concerned. Much time, chalk, 
and patience are lost when all or part of the class are working 
at the board. The inequality of time needed by various pu- 
pils, and the ease with which the brighter ones, out of a job 
temporarily, turn to things not connected with the subject, 
cause dismay to the teacher and an undesirable diversion 
for the rest of the class. 



42 Supervised Study in Mathematics and Science 

But when the teacher himself develops the work on the 
board, and the class follows the operations on paper, every 
pupil is of necessity alert and attentive, because he may be 
called upon at any time for some point. The entire class is 
therefore kept up to a high pitch of intensive work. The 
blackboard as a means of visualizing a demonstration before 
the whole class has a distinct value, but as a common working 
ground it is open to criticism. 

Summary on the Study of the Assignment. — As has already 
been stated, the various assignments should be placed upon 
the board before the meeting of the class. Let them be 
definite, concise. In the case of references, the exact title and 
page of the book referred to should be given. The use of the 
assignment sheets has been fully explained in Chapter One 
and need not be repeated here. 

Besides giving the pupils something definite to do at once, 
this first assignment of work will enable the teacher almost 
immediately to single out the poorer pupils and those who are 
likely to be the workers of maximum ability. Plans must be 
made at once to take care of both classes, — something to which 
the supervised study period is especially well adapted. A re- 
arrangement of the seating of the class will be made after a few 
days, as bad results sometimes come from the premature an- 
nouncement that permanent seats have been assigned. A 
workable plan is to put the slow workers on one side of the 
room, the rapid workers on the other side, and the rest of the 
class between them. If the class can be approximately so 
divided early in the term, no especial attention will be called 
to this classification, and the reasons therefor may remain a 
secret with the teacher. This arrangement will make it much 
easier to reach the two extremes of the class and give to them 



Divisions of Elementary Algebra 43 

special aid. It will also be a distinct help in administering the 
details of board work, individual instruction, and the use of 
supplementary material. 

Summary on the Silent Study. — The importance of getting 
to work at once without loss of time should be explained in the 
beginning of the term. The pupil should be made to feel that 
every minute is valuable, and that waste of time will not be 
countenanced. 

Various devices for keeping track of the completion of the 
different sections of the assignment will be given in succeeding 
lessons. The object of these sectional assignments is solely to 
aid the teacher in developing to the utmost each individual 
pupil in the class. This section of the supervised study period 
should be the most important part of the hour, because it is 
here that the pupil comes into personal contact with the teacher 
and receives first hand the kind of aid he needs. At the 
same time a certain amount of sympathetic relationship is 
developed on the part of both teacher and pupil. 

LESSON III 

UNIT OF INSTRUCTION I. — INTRODUCTION 

Lesson Type. — A How to Study Lesson 
Program or Time Schedule 

The Review 5 minutes 

The Assignment 30 minutes 

The Study of the Assignment 25 minutes 

Purpose. — The crux of the supervised or directed study 
period consists in the definite directions for the proper study 
of the subject in question, and careful explanations of just 
how to study rather than what to study. It is, therefore, 



44 Supervised Study in Mathematics and Science 

valuable during the course and especially at the beginning, 
that very definite rules for study should be outlined and in- 
sisted upon. 

The Review. — Subject Matter. A summary of the pre- 
vious day's assignment. 

Method. Call for questions on the exercises assigned. Ask 
for a few leading facts brought out by this assignment, such as : 

i. What was Newton's definition of algebra ? Comte's definition 
of arithmetic? ( Show their pictures to the class.) 

2. How did we get the word decimal? 

3. Mention some formulas used in arithmetic. 

4. What makes algebra a broader subject than arithmetic? 

Give a few examples like those in assignments I and II. 
Call on someone who completed the maximum assignment of 
yesterday's lesson to give the answers to the questions in this 
assignment. 

The Assignment. — 1. Methods of study. 

2. The system of supervised study explained by the teacher. 

3. The technic of the textbook in algebra. 

4. Definite instructions in how to study algebra. 
Methods of Study. — (a) External conditions. In the 

first place a correct physical environment is necessary. One 
must be in a comfortable seat, with good light coming over the 
left shoulder, breathing fresh air, and in a room of proper tem- 
perature. The pupil must have the necessary tools, as paper, 
well-sharpened pencil, ruler, textbook, etc. Next, the pupil 
must put himself in the proper attitude toward the subject 
and concentrate his mind upon his work. And then, with a 
determination and expectation to succeed, he is ready to com- 
mence his work. All these prerequisites will in time become 



Divisions of Elementary Algebra 45 

automatic if the teacher frequently calls the pupil's attention 
to them. It is easier to talk about concentration of mind than 
it is to achieve it, but the pupil should be carefully shown that 
the better control one has over his ability to keep his mind 
from wandering, the better student he will be and the sooner 
will he be able to master the task in hand. It takes will power 
and practice, but the teacher cannot emphasize too strongly 
the inestimable value of this acquirement, if once formed 
even to a limited degree. As McMurry suggests, 1 one of the 
best methods of acquiring concentration is through the 
employment of time tests, which require undivided attention. 
When we must do a certain thing within a definite time, we 
concentrate upon it. With sufficient practice, this may 
become habituated. 

(b) Technical factors. 1. The first technical factor in 
proper studying is the sensing of the problem. If the pupil 
simply goes at his work with a view of covering a certain 
amount of prescribed ground, without a realization of the 
problem involved, his study will degenerate into a mechanical 
grind. Every assignment, as stated before, should have some 
definite object or problem, around which the lesson will re- 
volve. In the first lesson, as outlined in these pages, it was 
"Is algebra practical?" In the second lesson it was "Is 
algebra entirely new?" and to-day it is " What is the best 
way in which to study algebra? " 

2. The second factor, after the recognition of the motive 
of the lesson, is bringing to bear upon it all our present knowl- 
edge of the problem and then supplementing the problem from 
our text and possibly other books and sources. 

3. We next seek to master the supplementary material by 

1 F. M. McMurry, "How to Study" ; Houghton Mifflin Co., 1909. 



46 Supervised Study in Mathematics and Science 

constantly referring to our present knowledge and associating 
our new facts with data already known. This may mean the 
employment of the memory, which, if rightly used, will be- 
come a means and not an end. Too much of our studying 
resolves itself into memorizing alone, and it then becomes a 
detriment. But as Miss Earhart well says, " Memory must 
not be substituted for thought but be based on thought." 1 

There are three kinds of memorizing: purely arbitrary 
memorizing, memorizing based on reasoning, and remember- 
ing the sequence rather than the things themselves. 

4. The fourth and last factor is the application of our data 
and material in the solving of our problems or in making our 
new power a part of ourselves. 

The Supervised Study Organization. A few words about 
the supervised study period may now be given. Each day's 
work will be divided into three parts : the review, the assign- 
ment, and study of the assignment. During the review, the 
previous day's work will be re-viewed in summarized form and 
any difficulties cleared up. This will usually take about 
15 minutes. The new work will be explained during the 
second time division, i.e. that of the assignment. This will 
take about 20 minutes but may vary in amount. The study 
of the new lesson will take place the last 25 minutes, and dur- 
ing this period the pupil will be expected to do as much of the 
new work as possible. Explain that he may feel free to raise 
his hand if he finds need of help, that he must not expect to 
have the teacher do his work, but only redirect him to find his 
own trouble, or lead him to see wherein his line of procedure 
is erroneous. The study period is not an opportunity for 

x Lida B. Earhart, "Teaching Children to Study"; Houghton Mifflin Co., 
1909. 



Divisions of Elementary Algebra 47 

getting someone else to do the pupil's task but an opportunity 
for getting proper directions so he may be able to do it 
himself. 

Explain that the assignment will be placed daily upon the 
board, that it will be in three parts, and that each pupil will 
be expected to do the first two parts and as many pupils as 
possible to do the third. The assignments should be so 
arranged that much of the work may be done during the period 
itself. The exact amount, of course, will depend on the pupil. 
It has been found in work of this kind that pupils are proud 
to excel their classmates. There are only the rarest instances 
of pupils deliberately doing only the minimum assignment. 

The Open Book. The pupils are asked to open the text- 
book in algebra while the teacher explains the structure. 
Comparison is made between title-page and the cover. A few 
words regarding the position or personality of the author, the 
name of the publisher, and the date of the book's publication 
may arouse some interest in the author as a second teacher 
of the class ; for, of course, the author of a textbook is to be 
regarded as a teacher. 

Have a pupil read the preface and then ask him a few 
questions which will bring out the reason for such an intro- 
duction to the book. 

Turn to the table of contents and note the various topics and 
subtopics of the subject. Compare this with the table of 
contents of some other text in algebra. Note the number of 
pages one or two units of instruction cover in your book and 
in some other text. If any topics are to be omitted from 
study, mention the fact. 

If your text has answers, give a few words as to their use 
and abuse. Teachers differ in their opinions regarding the 



48 Supervised Study in Mathematics and Science 

value of the printed answers, and, if it is impossible for the 
teacher to train the class to make them a side issue and not 
the most worn portion of the book, it is clear that their publi- 
cation is a serious mistake. 

Now open the book at the first page, calling attention at the 
same time to the fact that the paging of the book proper 
commences at this point. Announce that the work for the 
next day will begin here. 

This review of the make-up of the book proper may seem 
irrelevant to the study of algebra and a waste of time, but 
aside from the value of the general knowledge thus gleaned, 
it serves as an introduction to the text with which the pupils 
are to have intimate acquaintance during the year. It is 
well that they know something of the nature of the tool with 
which they are going to work. It often happens that things 
learned incidentally in connection with a subject will be of 
greater educational value than the subject matter itself. After 
all, our children are coming to school primarily to be educated 
and secondarily to learn algebra, Latin, or any other partic- 
ular subject. It is through these subjects that we hope to 
attain the ends of education. 1 

Instructions in How to Study. A few mimeographed 
directions may now be distributed to the class, and, after 
necessary explanations, the pupils may be told to insert them 
in their books for future reference. Explain that you may 
add other directions from time to time as the class progresses, 
and suggest that each pupil should feel free to make any 
suggestions for the enlargement of the list. 

1 " The Textbook — How to Use and Judge It" by Hall-Quest, The Macmillan 
Company, 19 18, gives a full discussion of what might well be covered in teach- 
ing pupils how to learn to use academic tools. 



Divisions of Elementary Algebra 49 

The list which follows is by no means perfect or complete ; 
it is simply suggestive : 

Suggestions for Effective Studying 

Be sure you understand the assignment. 

Study the meaning of the type of problem you are to solve, as 
suggested in its name, i.e. highest common factor, addition of 
radicals, etc. 

Recall the teacher's explanation of the new work. 

Study again the type form or example of the new problems. 

Understand thoroughly what is wanted before you begin to use 
your pencil. 

Avoid guesswork. 

Take time to think. Do not rush into an exercise trusting to 
luck you will strike it right. Be sure you are right ; then go ahead. 

Be sure you set the exercise down correctly on your paper. 

Work carefully ; it is easier to avoid mistakes than it is to find 
them. 

When you find a new application, study it until you master it. 
Expect each new problem to be different from the one preceding ; 
else, we would never advance. 

In case you cannot proceed, raise your hand. Do not expect 
the teacher to find your mistake but to direct you to find it yourself. 

Be neat in your work. A good workman is known by his neat 
performance. 

Slovenly habits of work lead to slovenly habits of thought. 

The Study of the Assignment. — The assignment for to- 
morrow will be in one section only. It will consist of some 
questions on the points of to-day's lesson, in regard to the 
attitude of study, factors of study, the technic of the text- 
book and the list of directions on how to study. These 
questions will help to focus the study on the essential features 
and to prevent wrong conclusions. A few sample questions 
are given : 

What is the proper temperature for a living room ? 
Why should the light come over the left shoulder ? 



50 Supervised Study in Mathematics and Science 

Suggest a good method of practice to attain concentration of 
mind. 

What was the problem of to-day's lesson in biology? 

Which of the three kinds of memorizing do you use in relating 
the incidents of a ball game ? 

Name some sources of supplementary material aside from the 
textbook. 

Is the preface necessary in every book ? 

Which do you think the author compiled first — the table of 
contents or the index? Give your reasons. 

Suggest any other directions than those given to you on the 
printed list. 

Which one of those given do you think would save you the most 
work, if carefully carried out ? 

BRING PAPER AND PENCIL TO-MORROW 

LESSON IV 

UNIT OF INSTRUCTION I. — INTRODUCTION 

Lesson Type. — An Inductive and How to Study 

Lesson 

Program or Time Schedule 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 

The Review. — Subject Matter. The questions on the 
previous lesson. 

Method. Write the various questions assigned yesterday 
upon slips of paper and have these in a loose pile, face down, 
upon the teacher's desk. Call on some pupil to come to the 
front of the room, to pick up one of the slips at random and, 
after reading it aloud, to proceed to answer it. If the class 
accepts this answer as correct and complete, ask someone 
else to repeat the process, and so on until all the questions have 



Divisions of Elementary Algebra 51 

been answered. If any question is not answered acceptably, 
replace the slip in the pile. This method of review will further 
help to break down the barrier between pupil and teacher, to 
accustom the pupils to talking before the class, to teach 
clearness and accuracy of expression, and to test the judgment 
of the value of the answer, thus giving all something to do. 
Furthermore it helps to review thoroughly the essential points 
of the preceding day's lesson. 

Note. — The lessons in algebra will not be based upon any special textbook, 
but the directions will be found applicable to any textbook upon the market. 
Copies of all the modern texts are upon the teacher's desk, and constant refer- 
ence is made to these either for supplementary examples or other material. 
Inasmuch as the present writer is interested chiefly in presenting a variety of 
schemes for teaching and training pupils in economical and effective methods 
of study, it is hoped that the point of view herein developed will be compre- 
hensive enough to include the situations that may arise in the use of any text in 
algebra. 

The Assignment. — 1. Instructions in how to study the 
printed page. 

2. Treatment of illustrative material. 

3. Treatment of class exercises (a) oral, (b) written. 
Instructions in How to Study. The pupils will open their 

texts and have their attention directed to the " definitions." 
It will be noted that this is the first unit of instruction as 
listed in the table of contents. (In the divisions of algebra 
outlined in Chapter Two, it is given as a unit of recitation 
under introduction; authors differ in the arrangement of the 
material.) 

The first paragraph is read carefully and the central or 
important point or problem discussed. The pupils will readily 
select the essential point of this paragraph. If there are any 
words which are new or not clearly understood, they should 



52 Supervised Study in Mathematics and Science 

be immediately defined by the teacher. Before we can com- 
prehend the sentence, we must know the meaning of its com- 
ponent parts. Explain the use of italics and heavier type. 
These take the place of the emphases in oral speech. These 
mechanical means call the attention of the reader to the im- 
portance of the word or phrase and should be specially noted 
by the pupil. 

If the sentence or paragraph is not clear at the first reading, 
reread it until the thought is mastered. Insist on the impor- 
tance of making reading thought-producing, and not simply 
a mechanical pronunciation of words. The language of 
mathematics is absolute and therefore cannot be read rapidly 
or slurringly ; every word means something. 

Now have someone reproduce the paragraph in his own 
words. Call upon a number of pupils to do the same thing, 
thus bringing out in various degrees of perfection the mean- 
ing of the assignment, and setting up a little rivalry for the 
best work. Emphasize the facts that we know what we 
can reproduce in our own words and that, when reproduced 
word for word like the text, we are thinking more of the 
mechanical reproduction than we are of the thought to be 
reproduced. 

The above outlined study of the paragraph might well be 
applied to the printed page of any book which is a subject of 
study, although some authorities strongly advise that the first 
reading of the page or section be made as a whole in order to 
get the general sense of the material. In algebra, however, 
since the textual matter is localized in its meaning, the pre- 
reading might be dispensed with. The reader will note also 
that the first three steps, mentioned in the preceding lesson 
as the order in which a subject should be studied, have been 



Divisions of Elementary Algebra 53 

followed, i.e. the point of view or problem, the data or mate- 
rial, and its mastery. Its application, as is often the case, 
will be made later. It often happens that we accumulate 
material through these steps for some length of time before 
we finally bring it together in the fourth or concluding step. 

Treatment of Illustrative Material. When we come to 
illustrative material, the example should be reworked on 
paper. Otherwise the pupil will mechanically read the oper- 
ation, think he understands it, and in a short time find that 
it has slipped away from his consciousness. This reworking 
of the example on paper will also help to fix it firmly in his 
mind and to establish each step thoroughly as it is written 
down, provided always that the pupil does the work with the 
motive of understanding the operations as they are evolved. 
Since this is an illustrative or model lesson, the teacher will 
also do the work which he will expect the pupils to do for 
themselves in their future study. 

For instance, suppose this formula is given : 

a = bXh. 

The meaning of the symbols is studied and then the value of 
the letters in a specific case is given and put upon the board. 

Thus, a equals 20 ; 

h equals 5. 

The question is, what is the value of J? The substitutions 
are now made in this formula and the solution performed. 
This operation should be repeated a number of times with 
varying values for the letters. 

Two things are being done in this operation : the pupil is 
learning how to interpret the printed word by his clarified 
perception, and he is also learning the fundamental character- 



54 Supervised Study in Mathematics and Science 

istics of algebra, the broad application of the algebraic func- 
tion. It might be well at this point to request some pupil to 
turn to the introduction of Milne's Standard Algebra 1 and 
read what that author has to say concerning it : 

The basis of algebra is found in arithmetic. Both arithmetic 
and algebra treat of number, and the student will find in algebra 
many things that were f amiliar to him in arithmetic. In fact, there 
is no clear line of demarcation between arithmetic and algebra. 
The fundamental principles of each are identical, but in algebra 
their application is broader than it is in arithmetic. 

The very attempt to make these principles universal leads to 
new kinds of numbers, and while the signs, symbols and definitions 
that are given in arithmetic appear in algebra, with their arithmet- 
ical meanings, yet in some instances they take on additional mean- 
ings. . . . 

In short, algebra affords a more general discussion of number and 
its laws than is found in arithmetic. 

Since with this introduction the pupil has an idea of the 
manner in which he should study, the teacher should further 
encourage him to proceed alone in his study. Questions need 
to be asked from time to time, however, to make sure that the 
pupil is following the directions and getting the right ideas. 
To illustrate, after the class has studied some paragraph or 
section, ample time having been allowed for the use of the dic- 
tionary, etc., ask some questions about it, and then call upon 
someone to state the problem involved ; someone else to restate 
it in his own words ; and others to supplement it from their 
own knowledge if possible. Thus we more and more throw 
the pupil upon his own resources but always with the proper 
methods of procedure before him, and careful supervision on 
the part of the teacher to see that he gets the correct interpreta- 
tion. He will eventually acquire the habit of study as outlined 

1 American Book Co., 1914. 



Divisions of Elementary Algebra 55 

above, which may be of more lasting value to him than the 
algebra itself. 

Treatment of Class Exercises, The treatment of exercises to 
be worked in class will differ somewhat from that of illustrative 
material. Suppose we wish to take up such exercises as the 
following : 

Read and explain : 

1. a+b. 3. aXb. 

2. a — b. 4. a + b. 

We have now accumulated our material and are ready for 
its application, or the fourth step. Here are definite examples 
of what we have been studying about up to this point. All 
study is for an end. As the final end of algebra is the solution 
of problems, so an intermediate step in the attainment of this 
end is the ability to perform the mechanical processes which 
will later be involved in their solution. 

Class exercises will therefore be of two kinds, (a) oral and 
(6) written. 

(a) Oral exercises. Some pupil is told to rise and read the 
first example. He is then asked to analyze or tell the meaning 
of it, which should be something like this : Two general mem- 
bers of different values are added together by indication, 
a added to b. The teacher should insist on complete answers, 
told in technical terms and in simple English. Clear thinking 
and clear expression will thus be unconsciously habituated 
by the pupil. 

(b) Written exercises. Exercises like the following, how- 
ever, may preferably be treated in a wholly different manner. 
Some such procedure as outlined here may be used or some 
modification of it : 



56 Supervised Study in Mathematics and Science 
If a = 3, 6 = 2, and c=%, find the value of each of the following: 



?a 2 a — b 

c a + b 

3. yfal 6. aX-. 

Most of the board work should be done by the teacher 
himself. The pupils should remain at their seats and either 
work on paper or tell the teacher what to write upon the 
board. This elimination of board work by the pupils will 
result in a more efficient use of the time of the period, as all 
members of the class will be either at work or on the lookout 
for possible questions. Every mark put upon the board should 
first be supplied by some member of the class and accepted by 
all as correct. Thus each member becomes personally inter- 
ested in the operations and alert to give directions or to detect 
errors. The class is thus kept up to a high tension and inten- 
sive work may be accomplished. The board work becomes 
a check and not a key, and the pupils feel that they have had 
a real part in its development. 

To illustrate, the work on the first example given above 
will proceed like this : the teacher will ask someone to read 
the example and to explain how to make the substitutions. He 
will then write it upon the board, directing the pupils to do like- 
wise on their papers. Another pupil will then be called upon 
to tell what is to be done next. As the pupil states the various 
steps, the teacher will place them upon the board, the class 
meanwhile doing the same on their papers. The pupil reciting 
will say something like this : The expression, 6a, means that 
the literal number a is taken 6 times, or that a multiplied by 



Divisions of Elementary Algebra 57 

6 constitutes the term in the numerator, and that the product 
is to be divided by b. Unless we give these literal numbers 
some values, the actual division can only be indicated as in 
the example. But if we assign some arbitrary values to the 
literals, we may substitute these values in the expression, 
perform the necessary operations, and reduce to its simplest 
term. In this case, since a is given the value 3 and b the value 
2, we find that 6a is equivalent to 18, and this divided by 2, or 
the value of 6, gives us the result, or 9. The work on the 
blackboard will appear as follows : 

6a 6X3 18 A 

— = ^= — = 9. Ans. 

b 2 2 

In this way, it would be well for the teacher to work on the 
board, with the assistance of the pupils, these six examples, in 
order that the pupils may learn how to handle written exer- 
cises. When written work is next required, it will be safe to 
assume that they will know how to go about their work, after 
one or two typical demonstrations have been given by the 
teacher. Work of this kind is oral or cooperative studying 
and should characterize every general assignment. 

The Study of the Assignment. — Assuming a list of 30 
graded exercises in the textbook in use, assign as 

I or Minimum Assignment. Exercises 1-20. 

// or Average Assignment. Exercises 21-30. 

Ill or Maximum Assignment. Exercises 15-20 on page 46 
of Ford and Ammerman's First Course in Algebra, 1 or exercises 
on page 7 of Slaught and Lennes' Elementary Algebra. 2 

Value of Outside Work. Many of the introductory lessons 
will be along the line of the foregoing, the majority of the 

1 The Macmillan Company. 2 Allyn and Bacon. 



58 Supervised Study in Mathematics and Science 

exercises being worked in the class under the supervision of 
the teacher. Each day a short assignment should be made, 
based on the ground covered and preferably taken from outside 
texts, especially the maximum assignment. This ought to be 
in the nature of a review of the work done in class and should 
be short enough to allow the majority of the class to complete 
all three assignments. These examples may be handed in the 
next day, but the best way for the teacher to make sure that 
the principles are thoroughly understood is to work out on the 
board through the minimum workers, or those who only 
completed the minimum assignment during the study period, 
a few typical examples during the review. 

The pupil must realize that the teacher is interested not so 
much in what the pupil has done as in what he can do now. 
// pupils could be made to know that work done outside of class 
is important only in so far as it makes them capable of doing 
something the next day in class, the incentive for getting other 
people to do their work would be greatly diminished. The out- 
side work must be insisted upon, unless the periods are long 
enough to have all the work done in class, but the credit 
should always be allowed largely upon the ability to do similar 
work again in class. It is the same rule that applies through 
the walks of life. The stenographer, the carpenter, the printer, 
the dentist, the worker of every sort is not paid for the record 
he has made in speed or the house he has built or the books 
he has printed or the bridge work he has done, but for his 
ability to do similar work again. To be sure, the experience has 
made him proficient, but we pay for results and not for the 
practice that has made the results possible. The one is 
indispensable, but the other is the criterion by which all of us 
are judged. 



Divisions of Elementary Algebra 59 

The next two or three lessons in the textbook may be 
worked out in a manner similar to the above. The amount 
of time spent on the work will of course depend on the 
book used and on the teacher. He may condense it into 
a shorter period or take even longer. The material given is 
merely suggestive and no teacher is expected to follow it 
verbatim. In fact, such a procedure would probably pre- 
determine the failure of supervised study, because more than 
anything else its successful operation depends on the origi- 
nality and individuality of the teacher himself. No system has 
been or ever will be evolved which automatically may be 
operated by someone and without change or adaptation be a 
success for everyone else. All that may be hoped for any 
method is that it be suggestive ; its final application and adap- 
tation, in the last analysis, lies with the teacher himself. In 
the words of Miss Simpson, author of a companion book in 
this series, "it is imperative that teachers adapt rather than 
adopt the methods suggested in these lessons." * 

LESSON V 

UNIT OF INSTRUCTION ILL— ADDITION 

Lesson Type. — An Inductive Lesson 
Program or Time Schedule 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 

The Review. — Upon the completion of each unit of in- 
struction, it is advisable to re-view the unit in toto. This may- 
take the form of an oral or written review. Various methods 
1 " Supervised Study in History "; The Macmillan Company. 



60 Supervised Study in Mathematics and Science 

should be used. In reviewing Positive and Negative Numbers, 
the following is suggested : 

Method. Write upon the board a large number of exercises 
covering the various phases of this topic, and call on different 
members of the class for the answers. These should be writ- 
ten upon the board in the proper place. Send a pupil to the 
board and as the answers are given have him write them down 
if he considers them correct. If he calls one correct when it is 
wrong, he must take his seat and another be sent to the board 
in his place. Thus the pupils are tested on their ability to 
solve the problems and, also, on their ability to judge correct 
results. It might be well to select someone to call on the dif- 
ferent pupils to recite, the teacher noting, however, that all 
or most of them are given a chance. When the pupil at the 
board makes a mistake, then the leader should take his place, 
and so on. Thus the review becomes socialized. It will 
provide interest for work that too often is needlessly weari- 
some. 

The Assignment. — i. The arithmetic preview. 

2. Recognition of the problem. 

3. Explanation of the type form. 

The Arithmetic Preview. As intimated in the second les- 
son (page 36), the pupils' present knowledge of arithmetic 
should always be drawn upon when possible to illustrate the 
new work. A few examples in adding numbers are followed by 
the implied addition of literal terms. The induction should 
be made by the class and the' Commutative Law of Addition 
deduced. After a few attempts, a workable law will be 
developed by some such questioning as this : how much is 
three and five ? five and three ? a and b ? b and a ? Ask 
whether it makes any difference in what order numbers or 



Divisions of Elementary Algebra 61 

letters are added. If the answer is " no," ask someone to 
state this principle in a sentence. Answer : Numbers can 
be added in any order. Tell the class that this is a law of 
order or the Commutative Law. 

Then broaden this principle when two or more numbers are 
grouped, as (5 plus 6) plus 4. What is the sum? (5 plus 4) 
plus 6 ? The sum is the same. Then we broaden the above 
law to include groups. Ask someone to state the revised 
principle. Answer : Numbers may be added in any order or 
group. Tell them that this is the Associative Law of Addition. 

Recognition of the Problem. Ask the pupils what is meant 
by a term, a monomial. The above illustrations are all 
monomials. Therefore the first problem under Addition will 
be addition of monomials, which becomes our first problem. 

Explanation of the Type Form. Place these examples upon 
the board : 

Add: 

1. 3 2. 3 boys 3. 3a 

_5_ 5 bQ y s .s* 

There will be no difficulty with the first two. Ask in 3, what a 
stands for. Someone will say " boys." But might it not 
stand for girls or houses or almost anything? The class 
will readily see that a may stand for anything and therefore 
the answer to the example will be 8a. 

Repeat with other simple monomials, all positive. Then 
put these examples on the board : 

Add: 

1. S 2. 5 dollars 3. 5a 

— 2_ — 2 dollars — 2a_ 

With their previous knowledge of positive and negative 
numbers, the class will see that in each case the coefficient is 3. 



62 Supervised Study in Mathematics and Science 

ASSIGNMENT AND STUDY SHEET 

Subject Elementary Algebra Period 2d 

Date September 7, 192 1 
Unit of Instruction Addition (III) 
Unit of Recitation Addition of monomials (I) 
Unit of Study Examples 1-25, text 
Lesson Type Inductive 



Review: 

Positive and Negative numbers. Exer- 
cises from Wheeler's Examples in Algebra, 
pages 4-7. 


Memoranda 
Examples on board. One writes 
answers which others give. Change 
when mistake is made. Work 
out laws. 


Assignment : 

1. Arithmetic preview. 

2. Recognition of new problems. 

3. Explain type form. 


What are monomials? 
Add : 3 3 boys 3a 
5 5 boys $a 


1. Minimum 
Exercises 1-20 


In text. 


2. Average 
Exercises 21-25 


In text. 


3. Maximum 

Exercises 43-50, Wells and Hart's New 
High School Algebra, page 38. 


Involve fractions and deci- 
mals. 


Study: 

See that the signs are correctly copied. 





Number of pupils solving minimum assignment 7 

Number of pupils solving average assignment 22 

Number of pupils solving maximum assignment 1 

Total "30 
Figure III 



Divisions of Elementary Algebra 63 

Now call on some pupil to stand and solve the first exam- 
ple, which may be : 

Add : 2a 

Similar exercises may be given orally. They may well be 
supplemented with others from the board until the principle 
is well understood. 

Then a typical example like the following may be developed 
on the board and the class set to work on the study of the new 
assignment : 

Add: 2X, §#, — x. 

The Study of the Assignment. — Assuming the textbook in 
use gives a list of 25 similar exercises, make the following 
assignments : 

J or Minimum Assignment. Exercises 1-20. 

II or Average Assignment. Exercises 21-25. 

III or Maximum Assignment. Exercises 43-50, page 38, 
in Wells and Hart's New High School Algebra. 1 These 
exercises are similar but a little more difficult, involving 
fractions and decimals. 

The Silent Study Period. — The Assignment Sheet. The 
division of the assignment into three parts, as suggested in 
Hall-Quest's book on Supervised Study, has been fully ex- 
plained in Chapter One of the present volume, which should be 
reread. The assignment numerals only should be used when 
designating the sections in placing the assignment upon the 
board and this should always be done before the class as- 
sembles. A sample sheet, made out to conform to this lesson, 
is given in full on page 62. It will aid the teacher materially 

1 D. C. Heath and Co. 



64 Supervised Study in Mathematics and Science 

if he will make these sheets out conscientiously during the 
term. There will then be no confusion or waste of time if 
additional exercises are needed during the class period. Many- 
precious moments are saved by a little foresight and planning. 
A lack of prearranged plans may also break down the morale 
of the class. Pupils are quick to respond to fine or poor 
executive ability when either is exhibited by the teacher. 
Napoleon was one of the world's greatest generals because he 
had the absolute confidence of every soldier under him. 
Teachers are generals of a little school army and the morale 
of the one is analogous to that of the other. 

The Completion of the Assignment, The minimum and 
average assignment should cover the amount of work that 
the teacher would ordinarily give under the old method of only 
one assignment for all. That is, the ordinary lesson for the 
next day would be about fourteen exercises in the text. But 
these have been broken up into two sections, the first of which 
should be worked by all within the 25 minute study period; 
if not, then the pupils who fail to complete this part need 
special attention. 

All the class is expected to have completed the average 
assignment before the next day and some pupils will do so 
before the end of the period. Those having trouble will 
have their work taken up during the review. 

The maximum assignment is designed primarily for the 
brighter and quicker pupils, those who are capable of doing 
more than the average amount of work. They should be 
given an opportunity of trying more difficult applications of 
the day's work. Not many will complete this part and it 
should not be demanded from all; but when done, some 
system of giving extra credit should be used. A good method 



Divisions of Elementary Algebra 65 

is to add half a credit to the monthly grade for every day that 
the maximum assignment was done correctly. In case a 
pupil did this correctly every day for a month, it would only 
mean ten extra credits, which might raise the grade from 80 to 
90. Very few would attain this maximum advance grade, 
however. But the teacher must be careful not to give too 
much credit to this advance work and so discourage the slower 
worker; it should be the aim always of the teacher to en- 
courage each one to do his best all the time. 

The Teacher's Duty during the Study Period. As soon as the 
study period begins, all start to work on the next day's assign- 
ment. The rate of speed will soon become uneven. Some 
will experience no difficulty and will advance rapidly ; 
others will be in trouble at the outset. For the latter, the up- 
raised hand will quickly bring the teacher with aid. Thus 
help comes when it is needed and at the time that the 
correct direction or word of helpful explanation will do the 
most good. The teacher must, of course, ever be on the alert 
to see that his help is corrective or suggestive and not simply 
a crutch. It should be directive and not simply finding the 
mistake for the pupil. The teacher, in quietly moving among 
the pupils, will note many wrong methods and incorrect habits 
of work which he can tactfully correct. Many small but 
important things, such as legible handwriting, neatness, care- 
ful arrangement, accuracy in copying the example, may be 
brought to the attention of the child at the time that he is 
working. It is a case of striking when the iron is hot. 

Occasionally a glaring error and its possible results may be 
called to the attention of the class ; for instance, the danger 
of mistaking a poorly formed 6 for a o if the loop is not care- 
fully attached to the bend of the figure below the top. Draw 



66 Supervised Study in Mathematics and Science 

the attention of the class to the fact that this little error 
invalidates the whole later process. This leads excellently to 
an explanation of the value of carefully checking the work as 
one proceeds. If each step is carefully gone over and checked 
for errors of omission or commission, before the next step is 
taken, valuable time may be saved. It is easier to avoid mis- 
takes than it is to find them. (See rule, page 49.) 

Verification. — A minute or so before the close of the period, 
check up the work done in class by the pupils. Various meth- 
ods may be employed, two or three of which will here be 
explained. Others will readily suggest themselves to the 
teacher. 
, First Method. Have some printed slips like this : 



Name- 



Class Period- 
Examples completed 

Assignment completed 



Time spent outside of class on to-day's 

lesson 

Teacher's check 



Figure IV 

Such a form can be filled out by the pupil in a minute's 
time ; the teacher can collect them and the next day file them 
with the papers handed in. It can then be noted how much 
of the lesson was done in class, and how that amount cor- 
responds with the examples done outside of the class period. 
This method will give the teacher the names of the pupils, 



Divisions of Elementary Algebra 67 

who failed to complete the minimum assignment in class, and 
these should have special attention the next day. This check- 
ing will also serve to tell the teacher whether his assignments 
are too long, too short, or about right. If the class does not 
approximately conform to the percentages mentioned in 
Chapter One, page 16, something is wrong in the assignment 
and an analysis of the situation should be made. Incidentally, 
this plan will take care of the roll call. 

Second Method. Have the pupils hand in all exercises com- 
pleted at the end of the period. Then the data may be 
computed by the teacher. The next day the examples worked 
outside of class may be handed in and filed with the others, 
which will thus constitute the completion of the assignment. 

Third Method. Just before the close of the period, call on 
all who have completed the minimum assignment to stand or 
raise their hands ; the teacher can either take down the names 
or have the pupils hand in their names on slips of paper. 
Then in like manner the names of those who have done the 
average and maximum assignments may be obtained. 

Fourth Method. Have the pupils check on their papers the 
point they had reached when the period terminated; then 
when these papers are handed in next day, the teacher may 
compile his own lists. 

LESSON VI 

UNIT OF INSTRUCTION m. — ADDITION 

Lesson Type. — An Inductive Lesson 

Program or Time Schedule 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 



68 Supervised Study in Mathematics and Science 

The division of the time of the class period, as stated at the 
beginning of each lesson, is that followed in the Canton High 
School, Canton, N. Y., where supervised study has been in 
operation since 191 5. The day is divided into five periods 
of one hour each. Longer periods, which would allow the 
pupil to do all his studying in school, would be ideal, but in 
many schools such a program could not be administered. If 
possible, the time schedule should be amplified, but the above 
proportion of time seems very adaptable, where different 
arrangements cannot be made. 

The Review. — Subject Matter. Addition of monomials. 

Method. The object of to-day's review is (a) to assist those 
pupils who had trouble with the assignment and (b) to give 
additional drill in this work to those who had no special 
difficulty but who were unable to complete the work. 

Those who have completed the threefold assignment and 
who have mastered the addition of monomials should] be 
allowed to proceed at once with the advance assignment in 
polynomials. This will serve as an inducement for intensive 
work and will encourage the brighter pupils to work harder 
and to solve all the exercises during the period if possible. 
This number will be small if the assignments have been care- 
fully planned. Those who have done part of the maximum 
assignment should be instructed to complete it. 

Now that we have the more advanced pupils working on 
the next lesson, or on the more difficult examples of the maxi- 
mum assignment, we can turn our attention to those who did 
not complete the minimum assignment or who had more or less 
difficulty. These pupils may be treated in different ways. If 
several failed on the same problem, they may be sent to the 
board to work on it under the supervision of the teacher. The 



Divisions of Elementary Algebra 69 

teacher can then watch their work and soon note the trouble. 
As soon as a pupil finishes one, he should commence on the 
next with which he experienced difficulty, and so on. This 
method is not advised, however, for reasons already stated 
against board work. 

A better method would be for the teacher himself to work 
out the problem, with the pupil directing him what to do, the 
others meanwhile following the process at their seats. 

Pupils should never be sent to the board, however, to work, 
for the benefit of others, examples which they solved them- 
selves. Board work has been inordinately stressed in mathe- 
matics. When a pupil can do a thing, he should not be asked 
to do it again ; it is his ability to do something which he could 
not do before which will make him advance. On the other 
hand, such exercises, written out on the board and afterwards 
read for the benefit of those who could not do them, will be of 
practically no value to them. Pupils can learn best by doing 
the work for themselves. 

As soon as the problems giving trouble have thus been 
solved, the remainder of the review period should be devoted to 
working additional ones of like nature, until this difficulty has 
been mastered. 

If there are still those who do not seem to be able to under- 
stand the problem, they should be given individual attention 
during the remainder of the period. The teacher must feel 
that his special task is to help the less capable ; children have 
varying degrees of ability and it is the peculiar province of the 
supervised study scheme that the backward ones are thus 
given special attention and brought up to the standard of the 
class as quickly as possible. The old process of the elimination 
of the dull pupil must give way to the new idea of reaching him 



70 Supervised Study in Mathematics and Science 

through a study of his particular difficulties and applying the 
proper stimulus which will enable him to " find himself." 

The Assignment. — i. Explanation of the method of add- 
ing polynomials. 

2. Recognition of the new problem and its attendant rule. 

Explanation of How to Add Polynomials. As in arithmetic 
we can only add or subtract terms of the same kind, so in 
algebra like must come under like, before we can add or sub- 
tract. In the example, Add: a +43/ and 2a — $y, we write it 

as follows : 

a+4y 

2a-5y 

30- y 

and add each term separately. Thus a plus 2a equals 3a, and 
4y plus minus $y equals minus y. If the order were different 
in the example, we would be obliged to rearrange the terms 
so that the a's would come under the a's, and the ;y's under the 
y's. 

Write another similar example on the board and ask some- 
one to direct the work. Put on another and send someone to 
the board to work it. If all claim to understand the opera- 
tions, pass on to the development of the problem involved in 
this lesson. 

Recognition of the New Problem and Its Rule. Ask what 
kind of expressions these are that the pupils have been manipu- 
lating. After you get the right term, polynomials, ask what 
is being done with them. Then ask someone to state the 
problem of to-day's lesson. The answer should be " Addition 
of Polynomials." 

It must not be forgotten that every day's lesson should 
have an object or problem; to-day it is Addition of Poly- 



Divisions of Elementary Algebra 71 

nomials. Various schemes may be used to emphasize it. 
One which has been used with success is writing it upon the 
board with yellow crayon. Thus it stands emblazoned in the 
mind of the child ; and, noticing it a number of times during 
the hour, he cannot forget that there is a definite object in 
view, and that the work in hand leads up to its understanding 
and solution. Again, if the special problem under consider- 
ation has not been thoroughly mastered, it remains upon the 
board and in this way the aim of the lesson is even more deeply 
imprinted upon the pupils' understanding. Pupils like to 
advance and if they find that another day must be spent upon 
some topic, — say addition of monomials — because they did 
not master it, renewed efforts will be made to move on to some- 
thing new. 

Develop the rule for adding polynomials by some such 
analytical method as this : ask why we put the various terms 
involving a in one column, and what kind of terms these are. 
Develop the definition of similar terms. When the example 
has been set down, what do we do? Draw a line and add each 
column, connecting them with their signs. The rule has thus 
been developed. Have some pupils state it in full. Repeat 
with different ones until you have something like this : 

Rule. — Arrange the similar terms in the same column, add 
each, and connect the resulting terms by their proper signs. 

Work two or three examples on the board, asking pupils 
to apply this rule by specific reference to the terms in the ex- 
ample thus solved. Such mechanical drill is necessary in all 
study of mathematics but, after all, there is only one object of 
drill, i.e. to grasp the principle involved, — and if by any means 
this may be done quickly, it ought in all justice to be employed. 
Mathematics should not become a master but a servant. 



72 Supervised Study in Mathematics and Science 

The Study of the Assignment. — I or Minimum Assignment. 
Exercises 2-23, in textbook. 

// or Average Assignment Exercises 24-27, in textbook. 

Ill or Maximum Assignment. Exercises 14-20, on page 41, 
DurelPs School Algebra. 1 

Verification. — Especial attention should be given to the 
pupils who were yesterday on the minimum list. For ex- 
pediency, the back of the assignment sheet used yesterday 
might be employed to record the names of these pupils. It 
might be a good plan, as soon after the organization of the 
class as possible, to reseat the pupils, placing those habitually in 
the minimum classification at the front of the room where 
they may be easily watched. Care should be taken, however, 
not to name the groups in such a way as to embarrass any 
pupil. The teacher will use tact under all circumstances. 
Certainly a gain in facility of class management should not 
be achieved at the loss resulting from humiliating or embar- 
rassing any member of the class. 

If the teacher is unable to locate a pupil's particular diffi- 
culty on account of illegible figures or general confusion of data, 
such a pupil may be sent to a side blackboard and there given 
a private lesson in some of the fundamentals of study. If he 
seems to have no conception of the problem, let him analyze 
it for the teacher, telling him what each term is and what it 
signifies. Make him comprehend the make-up of each term of 
the expression ; ask him why he puts it in a certain place, why 
he draws the line under it, how he treats signs in adding, etc. 

Such individual instruction takes time but is well worth 
while if thereby some boy or girl is saved from failure. One 
or two such private lessons like this each day, while the class 
1 Charles E. Merrill Co. 



Divisions of Elementary Algebra 73 

is at work upon the assignment, will serve to keep the teacher 
pretty busy, and yet the intensified effort will well repay the 
expenditures of time and trouble. The satisfaction of joy 
over seeing the weak pupil become strong is a great reward in 
itself. Such a case is like the physician's. It needs special 
study, painstaking oversight. But surely the restoration to 
health and the happy development of a " case " is a deep 
professional satisfaction. 

LESSON VII 

UNIT OF INSTRUCTION X. — THE EQUATION AND 
PROBLEMS 

Lesson Type. — An Expository and How to 
Study Lesson 
Program or Time Schedule 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 

The Review. — Subject Matter. Simple equations. 

Method. Give a speed test in reviewing simple equations. 
Have a large assortment of exercises on the board, or on 
mimeographed sheets ; see to it that the class is provided 
with paper and pencils ; set all at work on the minute. See 
how many exercises can be worked correctly in some definite 
length of time, say five minutes. When the time is up, have 
all stop immediately. Read the answers and ascertain how 
many attained a mark of 100. Collect all the papers and put 
the names of those having all correct upon the board. In 
looking over the papers which fall under 100, the teacher can 
note just where the trouble lies with the individual pupils and 
can remedy it the first chance he has. 



74 Supervised Study in Mathematics and Science 

Such a review, based on the time element, serves to 
strengthen the pupil's ability to concentrate, puts snap into 
the work, and lends an element of interest, as all young people 
like anything that savors of a contest. 

There are a number of excellent standardized algebra tests 
now on the market, which the teacher may use to advantage 
in this work. These tests were primarily constructed that 
there might be given an opportunity to teachers to compare 
the work done in their classes with that in other school systems. 
Since it is a fact that teachers will differ more or less markedly 
in their ordinary grading of examination papers and in their 
judgment of pupils' ability, the employment of tests, which 
have been used by a large number of teachers and the results 
of which have been standardized, gives an excellent means 
of evaluating the work in any class. But aside from this, 
these tests have other values for the teacher. They point out 
beyond doubt where weaknesses exist and allow a scientific 
basis for constructive work. Again, they arouse a vital inter- 
est in the results among the pupils tested because they like to 
know how their progress compares with that of other schools. 
The avidity with which pupils will strive to raise the standard 
of the school and of themselves offers the best inducement 
for intensified work in the classroom. 

(a) The Rugg and Clark tests. These consist of booklets, 
containing a series of sixteen tests on the various types of 
algebraic operations from one on collecting terms to one on 
quadratic equations. These may be given at one time at the 
completion of the work in algebra, but the author has found 
them of greater value in checking up his pupils on the com- 
pletion of each type process and noting wherein the pupils are 
weak, thus affording an opportunity for his diagnosing each 



Divisions of Elementary Algebra 75 

individual case and permitting him to give more drill in those 
processes that seem to need it. 

(b) The Hotz scales. These are in the form of sheets 
covering all the processes in algebra, including problems. The 
sheet on addition and subtraction gives examples in adding 
and subtracting terms, expressions, fractions, and radicals. 
The one on equations and formulas gives examples in simple 
equations, simultaneous equations, fractional, radical, and 
quadratic equations, and equations involving the manipula- 
tion of formulas. There are other sheets treating multiplica- 
tion and division, problems, and graphs. These tests are 
primarily useful in testing a class at the close of the work in 
algebra, as a means of comparison with standard scores. Used 
in connection with the Rugg and Clark tests, they form a 
valuable system of accurately and scientifically testing the 
progress of the class. 

The Assignment. — 1. Explanation of algebraic repre- 
sentation. 

2. The algebraic equations applied to a concrete problem. 

3. Definite rules for studying and solving problems. 

4. Analyses of several simple problems. 

The Representation of Concrete Things Algebraically. Some 
such questions as the following lead up very logically to the 
study of the problem by means of algebraic representation : 

1. Express the sum of five and three ; of a and b. 

2. Express the difference of five and three ; of a and b. 

3. What number increased by three is equal to eight ? 

4. What number diminished by three is equal to two ? 

5. How do the last two questions differ from the first two ? 

The pupils will see that in the last two questions something 
is lacking which is to be found. Tell them to indicate this 



76 Supervised Study in Mathematics and Science 

unknown by some letter, as x. Then the third question stated 
in terms of the known values and the unknown values, will 

read : Q 

x+3 = 8, 

and, after solving by transposition of terms, 
the fourth question will read : 

3-3=2, 

or, after solving, 

* = 5- 

6. If a pencil costs five cents, what will three cost ? 

7. If a pencil costs n cents, what will three cost ? 

8. Express the fact that a tablet costs five cents more than a 
pencil in both 6 and 7. 

9. Express the fact that two pencils and a tablet cost fifteen 
cents. Ans. 2^+5 = 15. 

10. Solve and find the cost of one pencil. Ans. w = S« 

These questions and others of like nature may be read to 
and be answered by the class; the result will be that the 
pupils will gradually sense the fact that by using literal num- 
bers, we are able to represent many things in a manner that 
we could not do otherwise. When definite values are assigned 
to letters, so that they will for the moment stand for some- 
thing concrete, they take on an entirely different meaning. 
Very strange "43 plus 53" may sound to a boy, but when we 
let x stands for dollars it assumes a very sensible and familiar 
form. Bring out the fact that the mechanical work preceding 
the study of problems has aimed at enabling the class to 
manipulate the resulting algebraic representation of some- 
thing concrete. 



Divisions of Elementary Algebra 77 

Application of the Equation to a Concrete Problem. Let us 
take this simple equation, x plus 5 equals 12. Have someone 
analyze it. It means that 5 added to some number unknown 
will give us 12. By the law of transposition of terms in equa- 
tions we solve, and x equals 7. 

Now suppose we have this problem : What number in- 
creased by 5 will be equal to 12 ? What are we trying to find? 
A certain number. Then since this is unknown, we will for 
the moment let x stand for it or equal it. How do we repre- 
sent increased value? By adding. Then how may we 
indicate the expression " number increased by 5"? Since 
x stands for the number, it will be x + $. But according to 
the remainder of the statement, it is equal to 12; then 
£+5 = 12. And solving, we find that x equals 7, or what 
we wanted to find. 

In like analytical manner take up several similar problems, 
such as: What number diminished by or increased by or 
exceeded by, etc., equals something? Lead the pupil in each 
case to see, through a prior arithmetical representation if 
necessary, the algebraic representation of the same. 

Definite Directions for Solving Problems. At this point 
either give the pupils the following definite steps, previously 
mimeographed, or have them written upon the board and 
copied by the pupils. 

a. Read the problem very carefully; study it until you know 
its every meaning. Close the book to see whether you can state 
it to yourself, silently. Then reopen the book to see whether you 
were right. 

b. Decide what is the thing wanted and represent it by x. 

e. If more than one unknown is involved, represent them by 
some other letters. 

d. Express in algebraic language each of the conditions men- 
tioned in the problem. 



78 Supervised Study in Mathematics and Science 

e. Make an equation of the two statements that express the 
same conditions. 

/. Solve for the unknown. 

g. There should be as many equations as there are unknown 
quantities. 

Illustration of the Directions. Given this problem : What 
number diminished by 8 is equal to 12? 

By a : We mentally analyze this to be : What number is there 
which will be equal to 12 or become 12 after 8 has been taken away ? 

By b : Number is the thing wanted ; therefore let x equal the 
number. 

By c : Only one unknown is implied in this problem. 

By d: x minus 8, and 12 are two expressions concerning the 
unknown. 

By e : They are equal, therefore, 

#—8=12. 
By /: #=20, or what was desired. 

Analyses of Several Problems. The teacher, through dif- 
ferent pupils, will then work out in similar analytical form, the 
analyses of several related problems. Insist on the above 
mentioned steps being followed each time; pupils must be 
taught how to study problems and not merely how to solve them. 
The pith of the whole thing lies in the ability of the pupil to 
read the problem intelligently and to understand it so thor- 
oughly that he can tell it in his own words. Pupils are apt to 
commence work before they fully comprehend what is given 
and what is wanted. 

The Study of the Assignment. — I or Minimum Assignment. 
Exercises 1-16, in text. (All of these should have been 
analyzed in class but not worked.) 

II or Average Assignment. Exercises 17-21, in text. (These 
have not been analyzed.) 



Divisions of Elementary Algebra 79 

III or Maximum Assignment. Exercises 26-30, page 176, 
Vosburgh and Gentleman's Junior High School Mathematics, 
Second Course. 

After this preliminary lesson on how to study problems, 
all of which should be very similar and not too difficult, it is 
advisable to take up problems in the following manner : 

Have each day a typical problem on the board ; as soon as 
the class assembles, let all read it over carefully and study 
it for a few minutes. Then call on someone to state what is 
wanted, someone else to state the expressions, someone to 
make the equation, and someone to solve it. Call on a number 
of different members to explain various phases of it, making 
the problem an object of class study and analysis. If the 
problem is simple in principle, it is often found profitable to 
have someone make up a similar problem. This method of 
having the pupil make his own problem and then solve it will 
be found an excellent means of getting him interested in this 
kind of work. Before the end of the year the problems which 
pupils will make up by themselves will astonish the most 
experienced teacher. Data may be supplied from current 
events, such as elections, ball games, business statistics, etc. 

The author cannot recommend too highly this method of 
spending each day a few minutes on a problem and then pro- 
ceeding with the regular work. It serves to keep the principles 
of solving problems ever before the class rather than for short, 
intermittent periods. Pupils do not tire of them but will 
really look forward to this phase of the day's work. Solving 
problems becomes a habit and what is quite generally con- 
sidered the hardest feature of algebra loses this aspect, be- 
cause the children have become so used to problem solving 
that it has become " second nature." Occasionally, an advance 



80 Supervised Study in Mathematics and Science 

lesson may be given on problems only ; but the above plan 
has been found, after careful trial, to cover their treatment 
adequately and well. 

LESSON VIII 

UNIT OF INSTRUCTION VII. — FACTORING 

Lesson Type. — A Socialized Lesson 

Program or Time Schedule 

The Review 30 minutes 

The Assignment 30 minutes 

The Review. — Subject Matter. Factoring ; all cases. 

Method. Have the boy and girl, who received the highest 
mark on the last grade card in algebra, choose sides. When 
this has been done, the lines should be placed as in the old 
fashioned " spelling-down bee." Commencing with the 
leaders and taking them alternately from the two sides, the 
teacher sends the pupils in turn to the board to work an 
exercise in factoring. Excellent material will be found in the 
numerous textbooks on the teacher's desk. If the exercise is 
worked correctly, the pupil returns to his place in the line and 
one from the opposite side goes to the board. If a pupil 
misses an exercise, he must take his seat and during the re- 
mainder of the contest work out all the exercises on paper and 
hand them in later to the teacher. 

In this way a large number and variety of exercises may be 
worked, and the pupils be tested for their skill in recognizing 
correct answers, for it is evident that the teacher should not 
take too active a part in reviews of this kind. The pupils 
know that they are expected to pass judgment on what is 
worked at the board. If serious confusion results, the 
teacher is resorted to as judge of the court of appeals. 



Divisions of Elementary Algebra 81 

When one side wins, i.e. has factored down its opponent, 
the assignment is taken up. 

The Assignment. — The following miscellaneous exercises 
in factoring, taken from various texts, will have been written 
upon the board. The first eleven will illustrate the different 
type forms of factoring, and a question or remark is set 
opposite each to direct the pupil in analyzing it. 

Factor : 

(Look out for a common factor.) 
(Difference of what? Rule?) 
(What type form have you here ?) 
(Be careful in your grouping.) 
(The parentheses indicate what?) 
(Is this a perfect square ?) 
(Note the powers ; odd or even?) 
(How may this be made a perfect square ?) 
(As a last resort, use factor theorem.) 
(How may such an example be best 
treated ?) 
11. z 3 *— a zb . (Keep in mind what 3^ and 3& are.) 

Then give the pupils an equal number of examples illus- 
trating all the various phases of factoring but in a different 
order, of course, and with no remarks. Tell the pupils to 
indicate, in addition to the answer, the type form which each 
illustrates, as : 

a z + b z = (a + b) (a 2 — ab+b 2 ). Ans. Sum of cubes. 
x n+1 +x=x(x n +i). Ans. Common term. 

Supply the pupils with copies of Wheeler's Algebra and 
refer them to page 70. Tell them to select all the exercises 
that illustrate some type form which is mentioned, such as 



1. 


3^-3^. 


2. 


a 3 — 1. 


3. 


X 2 + IJX + J2. 


4. 


ac — ax— 46c + $% 


5. 


(x+a) 2 —(x—a) 2 . 


6. 


4a 2 +4ab+b 2 . 


7. 


m b +n b . 


8. 


a*+W+aW. 


9. 


x? — 72+6. 


10. 


r 6 — s 6 . 



82 Supervised Study in Mathematics and Science 

the difference of squares, and note them on their papers 
without working them. For example, Nos. 3, 6, 10, etc. 

Then make a list of all the examples which illustrate some 
other case in factoring, and so on, covering all the principal 
cases. This may be carried out to any degree the teacher 
wishes, the idea being to acquaint the pupils thoroughly with 
the different type forms, to practice judgment in associating 
the example with its type form, and therefore in selecting the 
method which must be applied for its solution. 

LESSON IX 
UNIT OF INSTRUCTION IX. — FRACTIONS 

Lesson Type. — A Deductive and How to 
Study Lesson 

Program or Time Schedule 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 

The Review. — Subject Matter. Multiplication of fractions. 

Method. Write a number of exercises on the board, illus- 
trating the previous lesson on fractions. Send two pupils to 
the board, telling the others to work the examples on their 
papers. Then let the two at the board work the same example 
simultaneously. The one who solves it first may take his 
seat. As soon as the second pupil has solved it, the teacher 
sends a third pupil to the board to work the next exercise with 
him. In this way the second pupil, who had trouble, gets 
additional drill. If this method is continued the poorer ones 
will remain the longest and therefore get the most practice. 
Meanwhile all the others are busy. The pupils are expected 



Divisions of Elementary Algebra 83 

to solve the problems as rapidly as possible, the teacher during 
the meantime helping those at the seats who are experiencing 
difficulty. To vary this method, it is sometimes well to allow 
the one remaining at the board to choose his next opponent. 

The Assignment. — 1. Definitions of complex fractions. 

2. Directions for their solution. 

Complex Fractions Defined. A fraction containing one or 
more fractions in the numerator or denominator, is called a 
complex fraction. For example : 

x 



In other words, it means that the quotient obtained by 
dividing x by y is divided by the quotient obtained by dividing 
a by b. It may be set down like this : 

x m a 
y ' b 

and it then becomes similar to fractions in to-day's lesson. 
But sometimes the numerator of the complex fraction may 
itself be a mixed number or a series of fractions or another 
complex fraction, in which case it becomes necessary to follow 
out certain definite directions. These are : 

a. Simplify the numerator. 

b. Simplify the denominator. 

c. Divide the first result or quotient by the second. 
Illustration : 

'A 
«4 



84 Supervised Study in Mathematics and Science 

Bya: x+?-=^+l. 

J 3 3 

By&: •+J- 35 ?* 

J 3 3 

By.: V?±1^31±L or ^±Z + _A_ == ^+3!. AnSm 
3 3 3 3 a +b 3 a + b 

The Study of the Assignment. — I or Minimum Assign- 
ment. Exercises 2-1 1, in text. 

// or Average Assignment. Exercises 12-17, in text. 

Ill or Maximum Assignment. Make up and solve five 
complex fractions. 

The Silent Study. — The three steps as outlined above 
should be written upon the board with colored crayon so that 
the pupils may have them plainly in view. If they will care- 
fully follow out each step as applied to each exercise, the class 
will have no difficulty. If they do have trouble, it will be from 
carelessness. Most of the difficulty in fractions comes from 
the pupil's own illegible figures. The exercises on account of 
the awkward shape of their graphic representation are easily 
confused unless care be taken to set them down in good 
form and adhere to logical order in their solution. The 
teacher, by passing around among the pupils, will be able to 
note any such errors and he should avail himself of the op- 
portunity to correct them. 

In case someone has difficulty and calls upon the teacher for 
directions, unless the difficulty is easily found, it will be better 
for him to start anew, with the teacher overseeing that appli- 
cation is made of the three successive steps. If any help is 
given, it should be only to direct properly the application of 
these rules to the exercises under consideration. 



Divisions of Elementary Algebra 85 

LESSON X 

A Red Letter Day Program in the Nature of a 
Field Day 

I. Parade. — Each pupil in the class is to be assigned some 
rule covered in algebra during the first term, which he is to 
recite upon being called to the front of the room. For instance, 
the teacher, ox judge of the parade as he might be designated, 
will announce: To add two algebraic numbers. The pupil 
who has been assigned this rule will come forward and answer : 
11 If they have like signs, add the absolute values and prefix the 
common sign ; if they have unlike signs, find the difference of 
the absolute values and prefix the sign of the numerically 
greater.'' (Milne.) 

When all have been called on, the judge might award a 
prize, of no intrinsic value, to the pupil who made the best 
appearance and recited the rule in the most distinct voice. 

II. Races. — (Select three qualified pupils to act as judges.) 

1. Multiplication Race. 

Method. Have two exercises in multiplication, exactly alike, 
upon the board. Send two pupils to work on them ; the one 
getting his done first and correctly wins. 

2. Division Race. 

Method. Similar to above, but using different pupils. 

3. Championship Race in a Multiplication and Division 
Contest. 

Method. Give each of the winners in the first two races 
the same exercise, which will be a combination of multipli- 
cation and division. The one solving it correctly is considered 
the champion, and if deemed advisable may be awarded some 
prize, such as a colored ribbon. 



86 Supervised Study in Mathematics and Science 

4. Grand Relay Race in Removing Parentheses. 

Method. Put two exercises involving the removal of a 
number of signs of aggregation, upon the board. Pick out a 
relay team of at least as many pupils for a side as there will be 
complete operations. Start one from each side at the same 
instant and as soon as one operation is complete, let the next 
in order take his place. The side that first gets the exercise 
done correctly wins. It is suggested that two pupils be al- 
lowed to choose sides for this, members going to the board in 
the order chosen. 

III. Game of Factoring (modeled after baseball). 

Method. Select two teams of nine pupils each, preferably 
of pupils who have not taken part yet in the program, except 
the parade. These again may be chosen as noted above for 
the relay race. Also select an umpire. Each team will be 
composed of a pitcher, catcher, etc., as in a ball game. These 
positions will probably be best selected or assigned by the 
teacher, who will also explain the duties of the players and 
the rules of the game. 

The pitcher will read the exercises in factoring which will 
have been handed to him by the teacher. 

The catcher will try to tell the type form of the exercise before 
the batter can do so. 

The batter will tell the type form of the exercise as soon as 
he can. For instance, if the example is: a 2 +2ab+b 2 y he will 
say : " a perfect square.'' 

Each baseman and fielder will have been assigned some type 
form and his duty will be to solve the exercise by its application 
as soon as the batter refers it to him. For instance, the short- 
stop may have been assigned " a perfect square " as his position, 
so that as soon as the batter gave the exercise this classification, 



Divisions of Elementary Algebra 87 

the shortstop will solve it. If he correctly solves it, the 
batter is out ; if he cannot solve it, the batter makes a home 
run. 

If, however, the catcher gives the correct form before the 
batter does, and the fielder can solve it correctly the batter is 
also out ; if the fielder fails in this case, it is called a strike and 
the batter has another chance. Three such strikes will put 
him out. 

Again, if the batter gives the wrong type form, it counts 
a strike. Failure to understand the exercise at the first read- 
ing constitutes a foul ; the first two count as strikes, as in 
baseball. 

The game may be varied as to number of innings, accord- 
ing to the length of time that is available, but probably three 
will suffice for this program. 

As already mentioned, there should be an umpire to call 
strikes, fouls, etc. The teacher himself may act as referee 
in case of dispute. One or two score keepers may also be 
selected. 

IV. Picnic. — Method, Each of the following typical 
exercises in fractions may be considered to represent different 
articles of food, and the ability to solve them correctly will 
give the pupil a helping of each kind. Inability to solve the 
last one, for instance, which represents ice cream, would de- 
prive him of this dish. All pupils are given paper and told to 
solve the exercises which are placed upon the board. In 
parentheses is indicated what each exercise represents. The 
picnic may be held after school, if the period is not long 
enough, which probably will be the case. It would also be 
difficult to determine the amount of food needed before 
then. 



88 Supervised Study in Mathematics and Science 

Exercises 



5 I+S 

c-4 x 4-c 2 



— 2. 



Simplify : 
Simplify : 
Simplify : 

Simplify : 

Simplify : 

3^- 9 c-54 

iP 1 —- 6i "4~ i £ 

Reduce to mixed number : — * • 

i — i 



c + 2 i6 — c 2 

d s + 27 _ 1 _ d + s 
d s -27 ' (P + ^d + g' 

l + s 

ho 



l 2 + s 
h 2 o 



C'—C — 12 



(Sandwiches) 
(Cabbage salad) 
(Doughnuts) 

(Lemonade) 

(Cake) 
(Ice cream) 



LESSON XI 
UNIT OF INSTRUCTION XIV. — RADICALS 

Lesson Type. — A Socialized Review 
Program or Time Schedule 

The Review 6o minutes 

The Review. — Subject Matter. Reduction of radicals to 
the same order. 

Method. The idea of an occasional socialized lesson is to 
keep up the interest, throw the responsibility of failure upon 
the pupils, and impress them with the fact that algebra may be 
treated from the standpoint of practical application to the 
world's work. 



Divisions of Elementary Algebra 89 

The class is divided into two groups, one of which will con- 
stitute a sort of court and the remainder the witnesses. The 
teacher will select a judge, a lawyer, a jury of three and a court 
crier. They may be chosen on the basis of scholarship, the 
best pupil being judge, etc. 

The judge will take the teacher's chair. His business is to 
determine the fairness of procedure and to see that the rule 
of reducing radicals to the same order is carefully carried out. 
The lawyer is to question the witness on the exercise under 
consideration. The jurors are to decide on the correctness of 
the solution as given by the witnesses. The court crier is to 
assign the exercises to the various members of the class. The 
remaining pupils in the class constitute the witnesses who 
are to testify to their ability to solve the exercises which are 
under review. 

Procedure. The teacher swears in each officer by demand- 
ing his duty, thus giving all officials the opportunity to display 
their absolute grasp of the principles involved in the work. 

Answers to the following or similar questions might be 
required — 

Of the judge: 

" State the rule for reduction of radicals to the same order. " 

" Will you allow the lawyer to ask questions which might be 
misleading ?" 

" Will your attitude toward the witness be austere or sympa- 
thetic ?" 

" Will you keep order in the court room?" 

Of the lawyer: 

"What is your duty?" 

" Will you ask helpful questions or try to confuse the witness?" 

" Will you act on the principle that the witness is innocent of 



9<d Supervised Study in Mathematics and Science 

the charge that he cannot solve the exercise until he is proven 
guilty?" 

Of the jurors: 

" Will you promise to render a fair decision as you understand 
the rules of algebra relating to this subject?" 

Of the court crier: 

" Will you state the exercise in a clear and distinct voice?" 
" Will you be impartial in your assigning of exercises?" 

If all the above questions are answered satisfactorily, the 
officers take their respective stations and the court is declared 
open by the teacher and the program turned over to the judge, 
always subject, however, to recall by the teacher. 

The court crier calls on some pupil, who goes to the board 
and is given this exercise : 

Which is greater, V3 or \/fr? 

The witness then solves the exercise. The lawyer may ask 
him any question on his work or the method of solution, the 
object being to ascertain whether said witness thoroughly 
understands the exercise and is able to acquit himself of any 
inference as to guilt of the lack of such knowledge. The 
lawyer, for instance, might ask him why he cannot compare 
them as they stand, or what is the significance of the small 
figure 3. 

When the questions and the solution are completed, the 
jury passes judgment on the pupil's work with the word " cor- 
rect " or " incorrect " as the case may be If the verdict is 
" correct," another witness is called and the proceedings re- 
peated. If, however, the verdict is " incorrect," the judge 
must pass sentence. He may order a new trial immediately 
or he may impose some penalty, such as isolation in some part 



Divisions of Elementary Algebra 91 

of the room to work out at his seat some extra exercises se- 
lected by the teacher. When all the class have been examined 
in turn, the failing pupil, or pupils, may be given a new trial 
and thus offered an opportunity of redeeming his standing in 
the class. If he again proves unable to clear himself of the 
charge of " guilty of error," the judge may appoint someone to 
give him individual aid. The court may remain open until 
all the class have been cleared of any imputation of lack of 
ability to handle these exercises, or it may be terminated at 
any time by the teacher and court declared adjourned 
sine die. 

Many changes will occur to the teacher who tries out this 
plan, but the author believes that, aside from the added inter- 
est given to the solution of these exercises, the pupils are mean- 
while unconsciously learning invaluable lessons in court pro- 
cedure and social responsibility as well as lessons in self- 
expression and self-control. 

It is sometimes advisable to devote the entire period to this 
kind of work. Hence the assignment of exercises for further 
study will be given in one assortment only and it will be done 
outside of class and handed in the next day. In this case, 
twenty- five exercises to review further this type of problems 
may be assigned, taken from some supplementary textbook. 
In order to carry out the idea of the three assignments, the 
first ten might be considered a minimum number to be solved, 
the first fifteen the average number and the entire twenty- 
five the maximum quota. 

The Study of the Assignment. — Twenty-five similar 
exercises taken from Ford and Ammerman's First Course in 
Algebra, 1 page 256. 

1 The Macmillan Company. 



92 Supervised Study in Mathematics and Science 

LESSON XII 
UNIT OF INSTRUCTION XV. — QUADRATIC EQUATIONS 

Lesson Type. — An Expository and How to 
Study Lesson 

Program or Time Schedule 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 

The Review. — Subject Matter. Quadratic equations solved 
by factoring. 

Method. Have an example of this type written on each 
panel of the blackboard and send the minimum workers, as 
developed from the previous day's work, to solve them. Also 
place over them as monitors the maximum pupils, admonish- 
ing them not to solve the exercises, but to see that they are 
w r orked correctly. The teacher may meanwhile ask the 
other pupils necessary questions concerning such exercises. 
As soon as the exercises on the board have been completed and 
pronounced correct by the monitors, take up the new work, 
the work on the board being left for future inspection. 

The Assignment. — 1. Explanation that all quadratic equa- 
tions cannot be easily factored and that even those which 
can be factored may also be solved by other methods. 

2. Exposition of the method of completing the square. 

3. Statement and explanation of the rule for completing the 
square. 

Other Methods Sometimes Necessary. Write a quadratic 
equation upon the board which cannot be factored as it stands. 
For example : 

s 2 — 2#— 6=0. 



Divisions of Elementary Algebra 93 

Method of Completing the Square. Take this example : 

x 2 — x — 2 =0. 

It is in a form that can be factored at once into (x — 2) 
(x + i) ; but let us solve it as though it could not be so factored. 

a. First, transpose terms so that the unknowns will be on 
one side and the knowns on the other side of the equation. 

X 2 — X = 2. 

b. Then, unless, as in this case, the coefficient of the un- 
known to the second degree is unity, divide the equation 
through by the coefficient so that it will be unity. 

c. Now take one half the coefficient of the unknown to the 
first degree, in this case 1, and square it, i.e. § squared be- 
comes \. 

d. Add this to both sides of the equation, as 

X 2 — X+t = 2+|=f- 

e. Extract the square root of both members, and we have 

/. Solve for x\ x = 2 or — 1. Ans. 

g. Verify by substitution, 4 — 2 — 2=0, or 4 = 4, and 1 + 1 — 
2 =0, or 2 = 2. 

Statement of Rule for Completing the Square. This is al- 
ready given above, but divided into separate steps ; it may 
now be given entirely. 

Take a similar exercise and have someone tell what should 
be done, following each step by the directions above. Repeat 
until the class understands this new lesson. 

The Study of the Assignment. — I or Minimum Assign- 
ment. Exercises 3-20. 

II or Average Assignment. Exercises 21-25. 



94 Supervised Study in Mathematics and Science 

III or Maximum Assignment. Exercises 18, 19, 21, on 
page 269 of Ford and Ammerman's First Course in Algebra. 1 

The Silent Study. — Tell the pupils to apply each step as 
in the above set of rules, and to letter the result of each step 
with its corresponding letter. Their solutions should appear 
as follows : 



Exercise: 2x 2 +$x — 27=0. 

a. 2X 2 +$X = 2J 



(Transposition) 
a;2_|_3^ = !7. (Division by coefficient) 



C - Vs^ZV (Squaring) 

Uy —16' 

d , x a + M + J. = £2 + 9. = 2i6 + ^. (Addition) 

2 l6 2 l6 l6 l6 
2 l6 l6 

e. £+1==*=^- (Extracting square root) 

/. # =J r-, or - J ^- (Solving for x) 

£=3, or -4^. Ans. 

g ' 2i3)2 ti+ 9 = =l 7 7 : (Verification) 

If the pupil has trouble and it proves to be a matter of 
committing small errors of computation, let him go over his 
work again and carefully check each step. 

If, on the other hand, the pupil seems to have failed to 
grasp the principles involved and to be unable to apply the 
steps as suggested, let him go to the board and work there 
under the teacher's supervision. Let him read the rule over 
carefully and tell just what each step means. Let him study 
1 The Macmillan Company. 



Divisions of Elementary Algebra 95 

it until he does understand thoroughly just what it implies. 
A little direction from the teacher should relieve any mis- 
understanding of the method. 

If the pupil still fails to grasp the procedure, the teacher 
should solve the problem, explaining in detail each step as he 
proceeds. Then erase and have the pupil go over the same 
process, step by step. Many pupils fail to grasp thoroughly 
directions when given to a class at large, either because of 
inattention or failure to comprehend each step as it is de- 
veloped. This individual supervision will usually bring the 
desired results. When the pupil finds that you are not going 
to work his problem for him but will direct him how to do it, he 
will increase his efforts and try to master the technic. 

LESSON XIII 
TESTS 

Lesson Type. — An Examination 

Program or Time Schedule 

Examination 60 minutes 

Real Tests. — The daily review is in fact a continuous test 
of the ability of the pupil to do his work. As has already been 
emphasized, the pupil's outside work is important only in so 
far as it prepares him to do something similar again. After 
a pupil has worked thirty problems in addition of fractions, 
and handed them in to the teacher, and reported no difficulties, 
the teacher is certainly within his rights when he expects that 
pupil to be able to do similar examples under his observation 
in class. If he cannot, then his outside work is valueless and 
can be safely assumed not to be his own. The pupil should 
be made to realize that his assigned work is to make him pro- 



96 Supervised Study in Mathematics and Science 

ficient in a certain line and he must feel that the real test of 
his proficiency is not measured through his practice but 
through his ability to repeat the process. The review, there- 
fore, while being a part of the recitation participated in by all, 
should be especially directed toward the weaker pupils. 

Then the ability to understand the new work from day to day 
and to follow directions and get results is another real test of 
the pupil's advancement. It takes no written examination to 
demonstrate that a pupil is failing, when he continually comes 
to class unprepared and is unable to comprehend the new 
work as it is developed. In fact, the rapidity with which some 
grasp the new topic marks them specifically as unusual pupils 
in that line. 

Problems are in many ways real tests of the pupil's ability. 
The very statement of the problem in algebraic terms is 
indicative of logical reasoning; after this process becomes 
familiar, then the ability to solve problems so stated is further 
indicative of the pupil's mentality and mental growth. These 
processes characterize the whole field of this subject and 
readily and accurately measure the pupil's ability. Some 
pupils claim they can never master applied problems, and some 
teachers display a sympathetic attitude to this position by 
stating that, except for problems, their pupils could pass the 
final examination. While it is a fact that this is true in many 
cases, yet this is no reason why problems should be slighted ; 
in fact, it seems to the author almost an unanswerable argu- 
ment that problems should comprise the major part of the 
final examination. But there are problems and problems; 
all preposterous, ultra complex, and catch problems should, 
of course, be omitted. They should be fairly simple, fully 
reasonable, and straightforward in their applications. Some 



Divisions of Elementary Algebra 97 

texts, such as Ford and Ammerman/ have exceedingly well- 
selected, practical problems in accord with the best modern 
conceptions of instruction in algebra. The ability to handle 
such problems constitutes an almost ideal test of the pupils' 
knowledge of algebra. 

Besides these three daily records which themselves constitute 
real tests of the ability of the pupil to grasp the subject, there 
remains the formal written examination and its treatment. 

Written Tests. — 1. Object. The object of the written 
examination is twofold : (a) to find the points the pupils do 
not understand, in order to drill on them, and (6) to obtain 
a definite grade or per cent. The former is the really valuable 
thing ; the latter is immaterial. 

2. Testing for Weak Points. The teacher wishes to know 
definitely what points the class has mastered and what points 
have mastered the class. He is interested in the former only 
from an academic standpoint, but in the latter he is vitally 
interested because it is his business not only to diagnose but to 
treat and cure. 

Hence, after each unit of instruction has been completed, 
a review test is given to find out what the class has failed to 
grasp. These questions should be in the nature of drill ex- 
ercises, similar in nature to those already studied and given 
solely for the reason stated above. The writer has found it 
expedient and trustworthy to use some standardized test in 
which the results may be compared with a standard of excel- 
lency, rather than to use exercises of his own making. Rugg's 
tests 2 are exceedingly good for this work. There are others 
on the market, mentioned in Lesson VII. When the class 

1 "First and Second Course in Algebra" ; The Macmillan Company, 1919. 

2 University of Chicago Press. 



98 Supervised Study in Mathematics and Science 

fails to come up to the standard set in these tests, the 
pupils who fall below should be noted and special attention 
given to them on the principles in which they are found 
to be weak. Meanwhile those who have measured up to 
standard should be given more advanced work along the same 
line. 

The principle of the minimum-average-maximum classifi- 
cation should ever be kept in mind, and the teacher should 
strive not only to get all to qualify on the minimum require- 
ments but also should assist those who are capable to master 
more difficult applications and get higher grades. Teachers 
as well as pupils are very apt to be satisfied with the 
former. 

3. Testing for a Final Grade, The ideal method of giving 
grades is the average class mark. The written final examina- 
tion, however, is bound to be with us for some time to come, 
in at least some modified form. It should be along the line of 
questions which test reasoning ability and not the memory 
alone. As already noted, problems offer the most ideal method 
of formally testing the ability to " do algebra/' but questions 
of other kinds may be asked which will approximate, at 
least, the standard sought in the above statement. A sample 
examination of such nature follows, by way of illustration or 
suggestion, covering the work of the first term. 

A SUGGESTED EXAMINATION 

Directions : Answer all the questions under Part I. Necessary 
to pass, 3 correct from Part I and 2 correct from Part II. If all 
of Part I and Part II are correct, the grade will be 85%. If, in 
addition, 2 from Part III are correct, the pupil will receive honors. 



Divisions of Elementary Algebra 99 

Part I 

{Answer all the questions) 

1. Explain what you understand by positive and negative 
numbers, using some illustration, such as the thermometer, a bank 
account, etc. Give the rules and explanations of how to add, sub- 
tract, multiply, and divide signed terms. 

2. Make up and solve an example involving the removal of 
parentheses, containing at least twelve terms and at least three 
groups under signs of aggregation. 

3. Divide x b +y b by x+y giving reasons for each step, and 
prove your answer. 

4. Solve the following exercise in cancellation, stating what 
typical case in factoring is exemplified in each step, and explaining 
the process involved in each step : 

a 4 -5 4 ^ a 2 +b 2 
a 2 — 2ab+b 2 a 2 —ab 

5. Replace the question marks with values for x and y, and 
solve the finished example : 

? + ? 




((?+?)0 



Part II 

{Answer at least two) 

6. Simplify: a 3 -[a 2 -( 3 -a) - \2+a 2 -{i-a)+a?\], 

7. Factor: {a) 1-ra 8 ; {b) iox 2 ~7xy+y 2 ; {c) a*+b 2 y 2 +y*. 

a 2 
ac- 

8. Simplify : 



ac — 
4 



ac 

4 



ioo Supervised Study in Mathematics and Science 

Part III 
9. Expand: (.3^ — .4^ — 1.4) (.2X 2 +.2X — 2.5). 

10. Divide: a 2x+2 -a x+1 b A -2b 2x +3b x c x - 1 --c 2x - 2 by a x+l +b x -(f' 1 . 

11. Factor: a 3 — 7a +6. 

12. Simplify: - 4-i. 

1 a 

a 

1 

a 



a—x 



It will be noted that each part successively comprises 
exercises of an increasing difficulty, more technical and assum- 
ing advanced knowledge and ability on the part of the pupil. 

Value of This Form of Examinations. — This type of exam- 
inations seems to be a fairer test of the pupils' ability than 
the conventional one which comprises a series of exercises, 
any combination of which may be selected and the grade found 
by marking the paper. In such an examination there is really 
no testing of the ability of the brighter pupils to do superior 
work since all the questions are aimed at the average student. 
On the other hand, there is no minimum requirement for the 
pupils of lesser ability, which should be demanded of all. In 
other words, the pupils tested may attempt all and possibly 
salvage enough to get a passing grade, while their mastery of 
any one phase may be below standard. 

In the type above suggested, it is hoped that these malad- 
justments may be at least partially eliminated. In the first 
place, a certain number of exercises must be done correctly in 
order to receive a passing grade. These may be selected from 
Parts I and II, which give a choice in each part. If, therefore, 
the pupil cannot solve five eighths of these without error, he 



Divisions of Elementary Algebra 101 

will fail. If, in addition to these five, he succeeds in getting 
all of the first part and all of the second part correct, he will 
get a higher grade, or 85. And if there be anyone who can 
further work correctly two from the third part, he will receive 
honors. Thus each type of pupils, whether of minimum, 
average, or maximum ability, is given an opportunity of re- 
ceiving higher credit for additional work and work of a more 
advanced kind. But none can get honors by solving more 
of those of the simpler type or by carefully selecting all the 
easy exercises on the examination paper. So it would seem 
that this type of examinations definitely tests everyone in the 
class. 



SECOND SECTION 
PLANE GEOMETRY 



CHAPTER THREE 

DIVISIONS OF PLANE GEOMETRY 

Practically all texts in Plane Geometry agree on the 
Euclidian arrangement of material under the five-book 
system. These, with the introduction, thus become the units 
of instruction. 

A. Units of Instruction. 

I. Introduction. 

II. Book I. Rectilinear figures. 

III. Book II. The circle. 

IV. Book III. Proportion. Similar polygons. 
V. Book IV. Areas of polygons. 

VI. Book V. Regular polygons and circles. 

B. The Division of Each Unit into Units of Recitation. 

I. Introduction. The material in this unit will vary in 
different books but will probably comprise : 
Units of Recitation: 
i. Definitions. 

2. Axioms, postulates, etc. 

3. Oral exercises. 

4. Historical notes. 

II. Book I. Rectilinear Figures. 
Units of Recitation : 

1. Triangles. 

2. Parallel lines. 

3. Loci. 

4. Quadrilaterals. 

5. Polygons. 

6. Exercises and problems. 

105 



106 Supervised Study in Mathematics and Science 

III. Book II. Circles. 

Units of Recitation : 
i. Theorems on the circle. 

2. Problems on the circle. 

3. Exercises. 

IV. Book III. Proportion. Similar Polygons. 

Units of Recitation: 

1. Theorems on proportion. 

2. Similar polygons. 

3. Exercises and problems. 

V. Book IV. Areas of Polygons. 

Units of Recitation: 

1. Areas of equivalent and similar figures. 

2. Exercises and problems. 

VI. Book V. Regular Polygons and Measurement of 
Circles. 

Units of Recitation : 

1. Regular polygons. 

2. Measurement of the circle. 

3. Maxima and minima. 

4. Symmetry. 

5. Exercises and problems. 

LESSON I 
THE INSPIRATIONAL PREVIEW 

Meaning of Inspirational Preview. — As has already been 
emphasized in the companion introduction to algebra, the 
purpose of the preview is to arouse the child's desire to learn 
geometry. To this end it is important to skillfully advertise 
the subject. Such advertising should include flashes from the 
history of geometry, well proved values of its application and 



Divisions of Plane Geometry 107 

a brief survey of the contents of the course ; just enough of 
each to whet the appetite for more. The pupils are likely to 
feel that it is going to be dry or hard or futile, but such mis- 
conceptions may be quickly removed by a clear preview. 

History of Geometry. The word geometry — meaning in the 
Greek language, to measure land or earth — indicates that the 
science developed from the early practice of the modern 
science of surveying. It is not known with what people the 
science originated but certainly the Egyptians had acquired a 
considerable understanding of the subject as is attested by 
their pyramids, which are built in strictly geometric designs. 
Recently discovered tablets have proved that also the Babylo- 
nians were acquainted with this subject. 

But the first practical study of geometry for its own sake 
was made by the Greeks. Pythagoras, about 560 B.C., 
discovered many new propositions and added to the popularity 
and inspired increased study of the subject. Euclid was the 
first to make a successful attempt to write a book which would 
contain in an orderly manner all the known proofs, and so 
well did he do his work that all subsequent texts have been 
modeled after his book. He lived between 330 and 275 B.C. 

So we find that geometry, like algebra, is the combined 
product of many minds of many ages. Contributions are 
being made to its content at the present time. 

Practical Value. Geometry has a vital connection with 
many important phases of life. Indeed it is difficult to com- 
prehend how our modern civilization, with its machinery, 
buildings, bridges, ships, and other marvelous engineering 
accomplishments, could exist without the contributions which 
this science has made. Without a knowledge of geometry we 
would know nothing about the size of the earth, about our 



108 Supervised Study in Mathematics and Science 

solar system, about the universe as we to-day conceive it. 
The principles of geometry are used by engineers in construct- 
ing bridges, trestle work of all kinds, arches, etc. The em- 
ployment of formulas developed through use of geometry is 
universal in the application of mathematical knowledge to 
all kinds of mechanical construction. Such structures as the 
Brooklyn Bridge, the Eiffel Tower, the Capitol at Washington, 
the Ferris Wheel, the Roosevelt Dam, and countless others, 
are all the result of the application of geometric principles to 
practical engineering accomplishments. 

Again, geometry is made use of in designing mosaics, vault- 
ings, tile patterns, church windows, parquet flooring, steel 
ceilings, oilcloth, iron grilles, embroidery, lace work, etc. 

Although a study of mensuration begins in arithmetic, it 
is nevertheless only fair to say that its derivation is purely 
geometrical, and this science should be credited for its 
great contribution to this practical aspect of mathematics. 

If we believe in formal discipline, then geometry certainly 
deserves much credit for its contributions along this line. 
Plato believed explicitly in the mental value of this subject 
and it is said that he had a sign over his school of philosophy, 
reading, " He who knows not geometry may not enter here." 
Abraham Lincoln studied geometry to cultivate a logical 
mind. Geometry is practically the only subject in the school 
program which gives practice in the use of pure deductive logic. 
The concentration of mind and the method of logical steps re- 
quired to prove original problems in geometry combine to 
give one of the best mental exercises offered by any subject 
in school. The pupil learns to think clearly, logically, concisely, 
along mathematical lines. Its study offers more possibilities 
for correct and incisive thinking than any other branch of 



Divisions of Plane Geometry 109 

mathematics. If more of the method of geometrical proof 
were applied in situations demanding clear-cut thinking, it is 
not unlikely that many of the present-day issues would be 
better understood. 

A Bird's-eye View of the Course. Call the attention of the 
pupils to the fact that they have already had some geometry 
in arithmetic since they have studied about areas of plane and 
solid figures, such as rectangles, triangles, circles, cylinders, 
pyramids, spheres, and cones. In algebra, also, some of the 
facts of geometry have been employed in data stated in 
the problems. But now we come to a study of these 
truths from a new point of view, that of their value in 
themselves and not so much their application to the affairs of 
the world. 

Note with the class the division of the text into so-called 
books or chapters, each book devoted to some particular phase 
of the subject. Explain that these are the books of Euclid, 
and while they are now collected in one textbook, they still 
retain the old classification. Call the pupils' attention to 
the numerous drawings and devote a few words to an explana- 
tion of the value of neat, accurate figures. 

In conclusion and for a review of the above introductory 
work, put the following questions upon the board and require 
the answers to be written out and handed in the following 
day. Those given below are only suggestive ; better ones will 
no doubt occur to the teacher. 

1. The first syllable of geometry, or ge, means earth. Can 
you think of any other science which begins with ge? If so, of 
what is it the study? 

2. What natural phenomenon occurring regularly in Egypt 
caused the early development of surveying ? 



no Supervised Study in Mathematics and Science 

3. Find out, by reference to other texts in geometry, some 
other men than Pythagoras and Euclid who contributed to the 
development of this subject. 

4. State the Pythagorean theorem which you studied in arith- 
metic. 

5. How does geometry help us to a knowledge of the size of the 
earth ? Of what value is this knowledge to us ? 

6. Mention some decorative design you have noticed which 
is composed of geometric figures. 

7. What do you understand by deductive reasoning? 

8. You have learned in your arithmetic that the area of a 
rectangle is the product of the length and the width. How did the 
Egyptians make use of this rule? Has it any value in modern 
outdoor sports? 

9. Into how many books is the work in geometry divided ? 
10. Why must we have neatly drawn figures? Are they more 

necessary in geometry than in arithmetic ? Why ? 

LESSON II 

UNIT OF INSTRUCTION II. — Book I 
RECTILINEAR FIGURES 

Lesson Type. — A Deductive and How to Study 

Lesson 

Program or Time Schedule 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 

The Review. — Subject Matter. Properties of angles. 
Method. Draw a straight line upon the blackboard, as 

A B 



Divisions of Plane Geometry in 

Ask what a straight line is. Designate a point by C, as 

A . B 

C 

Ask what an angle is. Have someone go to the board and 
draw a number of angles, lettering each by ACB in which C 
is the vertex. Ask what kind of angle the above ACB is. 
What is the value in degrees of such an angle ? Draw a line 
to C from some other point outside, as P. 




What kind of angles are ACP and BCP? What is the 
value of the two? 

Extend PC through C to K. What kind of angles are 
PC A and KCA ? KCA and KCB ? 

What kind of line is PK? What kind of angle is PCK? 
Then what is the value in degrees of angle PCK? of PC A 
plus ACK? of PCB plus BCK? of ACK plus KCB? What 
kind of angles are they? If angle PC A is equal to ioo°, what 
is angle ACK equal to? Repeat with the various combina- 
tions, giving them different values. 

When the class has grasped the above, take up the assign- 
ment. 

The Assignment. — i. Statement of the problem. 

2. Deduction of its proof. 

3. Explanation of steps in proving a geometric proposition. 

4. Rules to govern the study of a proposition. 



ii2 Supervised Study in Mathematics and Science 

The Statement of the Problem. Explain that " proposition " 
is a general term for either a theorem or a problem. A theorem 
is a geometric proposition requiring proof ; a problem is a 
geometric proposition requiring construction. The propo- 
sition for to-day's consideration is in the form of a theorem, 
thus: 

Theorem. — // two straight lines intersect, the resulting 
vertical angles are equal. 

Deduction of the Proof. Using the figure already drawn by 
the teacher and referred to above, ask what vertical angles 
may be considered. Since there are two groups, we may study 
either pair, or ACK and PCB. 

Ask someone to state all the properties we know about the 
various angles involved in this figure. Have a pupil write 
these properties in algebraic form upon the board, as : 

I. PCA+PCB = i8o°. 

II. PCA+ACK = i8o°. 

III. ACK+KCB = 180°. 

IV. KCB + PCB = i8o°. 

What peculiar thing is noticeable about these equations? 
That they are all equal to 180 or the same thing. What can 
be said about things that are equal to the same thing? That 
they are equal to each other. Let us see if this will be of any 
value to us. 

What further significance can we note in the first two equa- 
tions besides the fact that they are both equal to 180 ? That 
each contains the same angle, PC A. What similar observa- 
tion can be made with reference to the second and third 
equations? That angle ACK is common. And in the 
third and fourth that the angle KCB is common. In our 



Divisions of Plane Geometry 113 

consideration of the above proposition, which of these three 
common angles might be of interest to us ? The class may 
select ACK. We shall take the second and third equations 

containing this angle, and combine them into a new equation 
since they are equal to each other : 

PCA+ACK = ACK+KCB. 

Have someone state why this is true. How can algebra be 
of sendee here? 

But since we have a common term on both sides of the equa- 
tion, what can we do with it? We can subtract this common 
angle from both sides without changing the value of the 
equation. Why ? Then we have : 

PCA = KCB. 

What kind of angles are these? Then we have proved 
that the two vertical angles are equal. But since these are 
not the two that we started out to prove equal, we will try 
another set of equations. 

Ask what angles we want left after solving the equation. 
ACK and PCB. Then let us take two equations which have 
these angles and also a common angle. The class will readily 
select the third and fourth, or 

ACK+ KCB =KCB + PCB. (Why?) 
ACK = PCB, (Why?) 

or what we wished to prove. 

Now, draw tw r o other intersecting lines upon the board, 
and call on someone to designate them by other letters, to 
find the various equations, to pick out the desired ones and 
prove the proposition. The pupil will tell what to do and the 
teacher will write the necessary operations on the board. 



ii4 Supervised Study in Mathematics and Science 

Then, have all the class draw two intersecting straight lines 
upon their papers, and go through the same process them- 
selves, calling you to their aid if they get into difficulty. The 
pupils will respond to this with avidity because all boys and 
girls like to achieve things themselves. 

Next, have them open their textbooks, which have been 
closed up to this time, and follow out the similar proof there 
developed. They will be very pleased to learn that they have 
already mastered the new lesson, and they have incidentally 
been given a lesson in how to study. 

An Explanation of the Steps to Take in Solving a Geometric 
Proposition. Explain that every demonstration of a geometric 
proposition is divided into definite, logical steps, as 

a. Statement of the theorem. 

b. Drawing of the figure. 

c. Stating data given in the theorem. 

d. Stating what is given to prove. 

e. The proof, consisting of steps and reasons. 

/. Conclusion (Q.E.D. or quod erat demonstrandum). 

Rules on How to Study a Proposition. The following set of 
rules may, with excellent results, be mimeographed and 
distributed to the class, as was suggested in algebra. 

Suggestions For Effective Studying 

Theorem. — a. Read and reread the theorem very carefully. 

b. Note what is given and what is wanted, 
v c. Review in your mind the properties of all geometric terms 
occurring therein. 

d. Have a blank card, the size of the printed page of your 
book ; call this card No. i. Have another card the length of the 
page but only one half as wide ; call this card No. 2. With card 



Divisions of Plane Geometry 115 

No. 1 cover up all the page except the theorem. Do not uncover 
any more of the page until you have mastered the above directions. 

Figure. — e. On blank paper make drawings to conform to the 
data given in the theorem. 

/. Push down the large card to disclose the drawing in the book 
and compare with yours. Do not put in any auxiliary lines until 
later ; letter your figure. 

Data. — g. Write on your paper under the head Data, the things 
you have stated concerning the theorem. Again push down the 
large card to compare. If your statement of data does not agree 
with the book, note wherein it differs and thoroughly understand it 
as given by the author before you proceed. 

To Prove. — h. Repeat this operation with the statement To 
Prove. Under this heading is given the thing desired by the theorem. 
(See suggestion b.) 

i. Divide the remainder of your paper into two equal parts by 
a vertical line. Label the first column Steps and the second column 
Reasons. 

Proof. — j. Slip down your leading card No. 1 to uncover the 
first step, keeping card No. 2 over the corresponding reason. 

k. If this statement incurs auxiliary lines, make them on your 
figure and compare for correctness with the figure in the book. 

I. Try to state a reason why this may be done. Then slip 
down card No. 2 to disclose the author's reason and note whether 
you were right. If not, master the correct reasoning for the opera- 
tion before proceeding. 

m. Repeat with the next step in the demonstration, disclosing 
first the step and then the reason for it after you have attempted to 
discover it for yourself. 

n. Repeat the process until you have completed the demonstra- 
tion. You will then have the complete work upon paper, also. 

0. Now close your book and after destroying your paper, try 
to write the complete demonstration upon new paper. When 
you are unable to proceed, refer to your text, but never memo- 
rize the steps, or you will never really understand geometry. 

In General. — p. Understand each step absolutely before proceed- 
ing any further. 



n6 Supervised Study in Mathematics and Science 

q. Take time. Thoroughly studying the theorem once should 
be enough. 

r. Always ask yourself why after each step. 

s. Number the steps and the reasons to correspond. 

/. Master the work from day to day ; do not let it master you. 
If you rely on memorization, you will become the slave instead of 
the master of this subject. 

The Study of the Assignment. — I or Minimum Assignment. 
Proposition I. 

// or Average Assignment. Exercises on Proposition I. 

Ill or Maximum Assignment. Prove the theorem in- 
formally by referring to Slaught and Lennes' Plane Geometry, 1 

PP- 33, 38, 79, 103. 

The Silent Study. — Pass from pupil to pupil to see that 
they are following out the directions as given above for the 
study of the lesson. Explain that a little practice with this 
method will soon make it automatic, and the pupil will find 
that before long he will master a demonstration in a short 
time. 

Insist on carefully drawn figures, neat and clearly lettered. 
Show the pupils how to make the cards suggested. Each 
one should be labeled and kept permanently in the book. 
They may be made of paper if desired; their value lies in 
keeping from the pupil's observation all except that which is 
being studied at the moment. Where they have been used, 
they have given good results. Their use should be encouraged 
by the teacher until the pupils realize their value as an aid 
to study. 

1 Allyn and Bacon. 



Divisions of Plane Geometry 117 

LESSON HI 

UNIT OF INSTRUCTION II. — Book I 
RECTILINEAR FIGURES 

Lesson Type. — A Deductive Lesson 

Program or Time Schedule 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 

The Review. — Subject Matter. Theorem. // two straight 
lines intersect, the vertical angles are equal. 

Method. Send two or three pupils, who completed the 
maximum assignment, to the board to write out the demonstra- 
tion of the theorem in full. Assign to those completing the 
average assignment the examples of the text to be worked 
upon the board. Review the demonstration of the theorem 
with the others. ,-v 

Have the figure already drawn upon the front board. Call 
on someone to state the theorem, someone else to give the data, 
and so on through the proof. When the demonstration has 
been completed in this way, have someone stand and go through 
the whole demonstration orally. Alter two or three have done 
this, erase the figure and have someone prove it, carrying the 
figure in his mind. This is good practice if carefully super- 
vised, for it keeps the minds of all concentrated on the de- 
velopment of the proof. Occasionally break in on the demon- 
stration to call on another to proceed with it. It is a good 
plan to have the class know that anyone is likely to be called 
upon at any time ; they will then give better attention to what 
is going on. ' - 

After the theorem has been thoroughly reviewed, ask for 



n8 Supervised Study in Mathematics and Science 

the answers to the examples of the assignment, noting those 
that are correct upon the board. Also have the pupils who 
completed the maximum assignment tell how the theorem 
was proved informally according to the references given. 
We are now ready to take up the new lesson. 

The Assignment. — i. Study of definitions. 

2. Explanation of the new theorem. 

The Study of Definitions of Triangles. Read the definitions 
over with the class, showing the pupils how to study them. 
Tell the pupils that the main points are to master each word as 
they read it ; to follow out all references to figures ; to look up 
the meaning of all the words they do not know, and the proper 
pronunciation of words they cannot pronounce. Then take up 
the new proposition. 

Explanation of the New Theorem. Follow the plan given in 
the preceding lesson. As new propositions occur from time to 
time, much of this work may be shortened as the pupils are 
able to do more of the deductive work themselves, following 
out the suggestions of the preceding rules. The teacher 
should determine, from day to day, what will probably con- 
stitute the real difficulties and should clear these up during 
the assignment period. It is better, however, to leave as 
much as possible to the silent study period and give those 
capable of solving the problems unaided a chance to do so. 
The class demonstration assumes that all are equally unable to 
study it out for themselves, which is not only not a fact but is 
stultifying to the more capable pupils. 

The Study of the Assignment. — I or Minimum Assignment. 
Proposition II on triangles. 

II or Average Assignment. Simple exercises on Proposi- 
tion II. 



Divisions of Plane Geometry 119 

III or Maximum Assignment. Examples on page 28, 
Wentworth and Smith's Plane Geometry. 1 

The Silent Study Period. — The above assignment having 
been placed in advance upon the board, the pupils are ready 
to begin their silent study of the lesson. Tell them to open 
their books at the new proposition, and to study it according 
to the directions given them yesterday. With the large card 
over all the page except the theorem, they should begin with 
rule a. For a day or two, or until they have acquired the 
correct method of studying the demonstration as given in the 
book, it will be best for the teacher to direct orally this 
study, guiding the pupils to apply correctly these direc- 
tions. Since one purpose of supervised study is to develop 
eventually in the pupil a knowledge of how to study, 
too much emphasis cannot be placed upon this practice. 
In time, if such study is directed and insisted upon, the pupils 
will find it unnecessary to rely upon the teacher's guidance 
and will be able to study the lesson unaided. This should 
be, of course, the ideal sought for, but it will take more 
or less time and the acquisition will only come through 
patience and perseverance. 

After the pupils have been given time to digest the stated 
theorem, ask someone to state what is given and what is 
wanted. Not until it is clearly understood that two triangles 
are given, each of which has two sides of one equal to two sides 
of the other, and the included angle of the one equal to the 
corresponding included angle of the other, and that we desire 
to prove that under these conditions the two triangles are 
equal, are we ready to pass to the third rule. Call on various 
pupils to state these things. A clear comprehension of these 

1 Ginn and Co, 



120 Supervised Study in Mathematics and Science 

two elements of every proposition is absolutely necessary 
before we proceed. 

Now ask what we know concerning the properties of the 
geometric terms involved in this proposition. Bring out that 
all triangles have three sides ; that there are also three angles 
and that each angle is included between two sides. Such an 
angle is called an included angle. Why? Ask when geo- 
metric figures may be said to be equal. If the pupils are 
unable to tell, refer them to the axiom. Having now analyzed 
all of the properties, turn to rule d. While they keep the large 
card over all of the page except the statement of the propo- 
sition, they will next draw figures upon their papers to con- 
form to the facts of the theorem. When this has been done, 
tell them to lower the card to disclose the figure and to compare, 
lettering it to conform with the lettering of the drawing in the 
textbook. 

Next, tell them to write after the heading Given the facts 
that have been stated in the theorem as data. Before the 
card is pushed down again, ask someone to tell you what he 
has written under this head. Discuss it with the class to see 
if it is complete. Then tell them to uncover this section of the 
book to compare with the statement of the author. 

Repeat the process with the heading To Prove. In order 
to expedite the work, the teacher may discuss each of these 
points orally rather than have the pupils write them down, 
possibly incorrectly. The thing desired is the deduction of 
the various steps, as far as possible, by the pupils themselves 
without referring to the book, except for verification. 

The same procedure may be followed throughout the study 
of the demonstration. The use of the divided card, as ex- 
plained in Lesson II, will tend to make the pupil study each 



Divisions of Plane Geometry 121 

step out himself before referring to the author. This should 
especially be insisted upon in giving the reasons. 

After the first step in the proof has been studied, ask the 
pupils, preferably individually, to give some reason why this 
step has been taken. The teacher will thus get all the class to 
thinking, and they will review mentally all the facts which they 
have learned up to this time. This is a point which cannot be 
overemphasized. Ability to solve originals or in fact to do any 
part of the work in geometry requires a continual revolving 
in the mind of all our previous knowledge with a view of 
applying it to the specific case under consideration. The 
difficulty with originals often results from this very inability 
or failure to practice. 

When the complete demonstration has been studied in this 
way, the teacher may tell the class to take a sheet of paper and 
try to rewrite the proof, or better, to reconstruct the proof. 
The only thing which will at all necessitate the use of the 
memory will be the order of the steps and the continual review 
mentally of previously learned facts concerning geometric 
terms. 

As the pupils begin this work, the teacher will pass among 
them to note any special difficulties. Occasionally a leading 
question will set the pupil aright but definite answers should 
be avoided ; it is our duty to lead the pupil to see his trouble 
rather than simply to find it for him. 

The Study of the Advance Assignments. — As soon as a 
pupil has mastered the new proposition, he should go to work 
upon the second assignment. It will be best at first to demand 
that the answers to these exercises be written out. It will 
take more time and later may be dispensed with, but at first 
it will cause the pupil to concentrate his attention more 



122 Supervised Study in Mathematics and Science 

definitely upon his work. These first exercises are simple and 
will not take much time or paper, but the pupil by this method 
will soon acquire the habit of being accurate, comprehensive, 
and neat. 

In case there are any exercises that might involve special 
difficulty, a leading question concerning it may be written 
upon the board, numbered to refer definitely to such 
exercises. 

The examples assigned for the maximum assignment may 
either be written upon the board or the pupil allowed to take 
the book referred to. In case more than one should reach 
these examples during the period, the difficulty arising from 
having only one copy would be eliminated by the use of the 
board. On the other hand, the actual use of supplementary 
textbooks by the pupil is an excellent practice and, whenever 
it is feasible, such additional books should be provided and 
in as large a variety as possible. 

Encourage the pupils who are capable to attempt the 
maximum assignment but not to the detriment of their other 
work. Make it clear, too, that it is only to be studied after 
the mastery of the first and second parts. Its purpose is to 
keep employed those in the class who are especially capable and 
thus further to develop their powers. As a rule, these ex- 
amples should be of a more difficult nature. This method, 
carried out to its ideal administration, would permit these 
pupils to proceed with new propositions, and so finish the 
book ahead of the others ; but this involves too much difficulty 
for the average school to attempt and should only be done 
under exceptional circumstances. For a further discussion 
of this important phase of supervised study as relating to 
higher mathematics ; see Chapter Four, 



Divisions of Plane Geometry 123 

LESSON IV 

UNIT OF INSTRUCTION H. — Book I 
RECTILINEAR FIGURES 

Lesson Type. — A How to Study Lesson 

Program or Time Schedule 

The Review 15 minutes 

The Assignment . 20 minutes 

The Study of the Assignment 25 minutes 

The Review. — Subject Matter. The first three propositions 
of Book I. 

Method. Send three pupils to the board to draw the figures 
of the propositions which are to be reviewed. While these are 
being put on the board, quickly review the class on the leading 
facts already studied in regard to angles, triangles, etc Such 
questions as the following are suggestive : 

What kind of angles are equal ? 

State under what two conditions, already studied, triangles are 
equal. 

What method of proof is employed to prove triangles equal ? 
When is a line perpendicular to another line ? 
What do you mean by bisecting a line ? an angle ? 

The questions asked should always review the knowledge 
acquired recently \ and facts that might be useful in studying 
new propositions and originals should be gradually introduced 
by this means. 

The figures having been drawn upon the board, call on some 
pupil to state the theorem about vertical angles and to prove 
it, using the figure on the board. See that he omits nothing 
important. It is best to prompt the one reciting as little as 
possible, because, if the pupil learns to depend on the teacher 



124 Supervised Study in Mathematics and Science 

for his approval or disapproval of every statement, he will 
never be able alone to complete a demonstration. The teacher 
must avoid aiding him too much ; if the instructor breaks in 
with " why " every time the pupil fails to give a reason, he 
will early learn to expect it and will not trouble himself to give 
it unaided. A better way is to stop him when he makes an error 
and let someone else proceed with the proof. He will soon 
learn that he must rely on himself, and will exert himself to 
be thorough. After another pupil has completed the demon- 
stration successfully, tell the one who failed, unless he himself 
senses it, wherein he made his mistake, and let him try again. 

Proved orally, many demonstrations can be covered in the 
allotted time. As soon as this proposition is given correctly 
and the pupil who failed has been able without help to give 
it, call on someone to prove the second proposition, and so on 
until the subject matter of the review has been covered. It is 
well to review each day not only the propositions which were 
the immediate subject of study for that day, but also to review 
continually others, dwelling especially on those which have 
given the most difficulty. In cases of complicated figures, let 
the pupil go to the board and use the pointer, and in the case 
of many equations being necessary, it will help the pupil to 
allow him to write them on the board. 

If there are some in the class who still have difficulty and are 
unable to go through a demonstration correctly, entirely alone, 
let them go to the board during the study period and write it 
out. This method, however, although much used, is of special 
value only in the case of pupils who are unable to prove the 
proposition orally before the class. It ordinarily takes too 
much time to be of much worth. 

When a pupil is demonstrating a proposition, the teacher 



Divisions of Plane Geometry 125 

must bear in mind that if the time thus spent is not to be 
wasted by the rest of the class, the pupil must talk loud enough 
for all to hear him clearly. Let him stand at one side of the 
figure and talk to the class. The pupil should be made to feel 
that he is taking their time and they are entitled to all the bene- 
fits that may accrue to them from his work. 

A few minutes might also be spent in running over the 
exercises, which are based on the day's lesson. These may be 
given orally or written upon the board, depending on their 
nature and their value. When possible, similar exercises with 
different values may well be given by the teacher ; and the 
pupils may be encouraged to make up others of a like nature. 

The Assignment. — The new lesson will be on originals ; and 
since it is the first lesson on these, a few words as to how to 
study them properly will not be out of place. 

Notebooks. The author advises the use of notebooks ; one 
to use in class for original exercises and to be handed over to the 
teacher at the close of the period ; the other for use outside of 
class for additional exercises which were not done under the 
immediate observation of the teacher. The former will 
contain beyond peradventure the pupil's own work ; the latter 
may be assumed to be such and assessed according to the 
ability of the pupil to work similar examples in class. 

The pupils are told to open their textbooks and to read the 
first exercise. Opening their notebooks, they will draw the 
figure, state what is given, and what is to be found. Ask 
various questions, taking pains to see that all understand the 
data given and the results desired. The questions that you 
will ask to-day will illustrate how the pupil is later to question 
himself when studying similar problems. The following 
illustrations may serve to make the method clear. 



126 Supervised Study in Mathematics and Science 

Example. Given lines AB and CD bisecting each other at 0. 
Draw straight lines connecting A and C, and D and B. 
Prove that AACO = AOBD. 

Questions. When lines bisect each other, what results? Which 
segments, then, are made equal in this example ? 

To prove that triangles are equal, we must try them by three 
conditions ; what are they ? 

In this figure, what do we know about each triangle ? 

Does this give us enough knowledge of each to throw them into 
one of the conditions under which triangles may be proved equal ? 

Then, since we know that each triangle has two sides and the 
included angle equal respectively to two sides and the included 
angle of the other, what must follow? 

Is this what we wish to prove ? 

Solution. Tell the pupils to write out later the entire proof 
according to the method of the demonstrations in the textbook, 
using the divided page, with the steps on the left of the line and 
the reasons on the right. 

Example, Given Z ABC bisected by BY, P is any point in BY, 
equal lines are dropped from P to the sides of the angle, as PM 
and PK. Prove that ABPM = ABPK. 

Questions. First ask the pupils what questions they think they 
should ask themselves concerning this example. Naturally many 
questions will come to the mind of the individual pupil which will 
be found to have no bearing on the problem under consideration, 
but it will be best to exhaust all possible conditions even at the 
risk of some matter which is not pertinent. He will learn that in 
the investigation of any new subject, much of the effort exerted 
fails to be of consequence, but it is necessary to bring to bear all 
the known facts so that we may study their relationships. 

Explain that, in the first place, the pupils must study the 
examples by passing over in their minds all possible related 
facts, and, in the second place, by eliminating all except those 
that will bear directly on the problem involved. We know 
such and such a thing. What ought we to know if we solve 



Divisions of Plane Geometry 127 

the problem? Do these facts help us and how? If not, are 
there any others that we have omitted that we could possibly 
use? 

In this way analyze five or six examples in to-morrow's 
assignment, and then require the pupils to write out in full in 
their notebooks the demonstration of each in the way pre- 
viously indicated. 

The Value of Definite Rules for the Study of Different Phases 
of the Subject. Give each pupil, on mimeographed sheets if 
possible, the following rules for attacking and solving original 
exercises. If he will adhere rigidly and conscientiously to these 
directions he will be able to solve any original exercise he may 
meet. And not only that, he will save himself the time that 
is often spent in aimless study — study that brings to bear no 
intelligent and directed effort. It is comparable to the way 
the expert machinist looks for " trouble " and the way the 
unskilled layman looks for it. The expert does not aimlessly 
take off nuts, loosen joints, and dislocate couplings ; before 
he touches a thing, he studies his problem. How should the 
machine function; and, is it so functioning? If not, what 
might cause such trouble? And so on, until he finally elimi- 
nates many improbable or impossible causes and reduces it to 
something which is probable or possible, and then he goes 
after that thing. The layman, on the other hand, not studying 
it out beforehand, will do something here and something there, 
until the chances are he will add to his trouble instead of re- 
moving it. So in solving exercises in geometry, a little care- 
ful analyzing of the data, the problems involved, the things 
known and the things desired, and how one may affect the 
other, will lead eventually and logically to the correct 
solution. 



128 Supervised Study in Mathematics and Science 

Suggestions for Studying Originals 

a. Digest every word in the problem. 

b. Make the figure carefully and go over it to see that it follows 
directions. 

c. Ask yourself what you know about every word in the data. 

d. Ask yourself how this knowledge may be applied to the 
question under consideration. 

e. State carefully what is to be proved. 

/. Review mentally under what conditions the proof may be made. 
g. Write out the proof in full, giving a reason for each step. 

The Study of the Assignment. — I or Minimum Assignment. 
Exercises 1-6, in text. 

II or Average Assignment. Exercises 7-9, in text. 

III or Maximum Assignment. Exercises 10 and 11, in text. 
Method of Manipulating the Notebooks. As noted above, 

have all the pupils write out in their notebook, or Notebook A , 
all the exercises worked during the class period ; work in the 
other notebook, or Notebook B, those done outside of class. 
At the close of the period, stand at the door and collect the 
former notebooks. It will take only a few minutes to look 
them over and check those found to be incorrect. Make a 
note of these failing pupils on the back of the day's assign- 
ment sheet, and take up the unsolved problems with these 
pupils the next day. 

The notebook is easily kept in order, it indicates the pupil's 
work and progress from day to day and becomes an efficient 
reminder of poor work. Such remarks as " too slovenly," 
" don't guess," " figures inaccurate," etc., might well call the 
attention of the pupil to the reason for his failures. Rubber 
stamps with such phrases could be used to advantage with a 
view of saving time and energy. Written work taken up but 
never returned with criticism is inefficient and does not repay 



Divisions of Plane Geometry 129 

the teacher for the time he spends looking it over. Papers 
handed in and simply marked with a grade or per cent are of 
little help to teacher or pupil. The real benefits are derived 
only when the errors are noted and the attention of the pupil 
called to them so that he may avoid making similar mistakes. 

The notebooks containing the work done outside of the 
class period may be looked over during the study of the assign- 
ment and errors brought at once to the notice of the pupil. 
They may be returned to the pupils at the close of the period, 
or collected only as the teacher is able to look them over. 
Hence the value of notebooks ; they are always ready for 
inspection, and their administration is very easily effected. 

All pupils ought not to be required to do the same amount 
of original work ; all, however, should do the minimum assign- 
ment, which should cover all the simpler applications. The 
more efficient workers should be encouraged to solve more 
difficult exercises, and so all may be kept up to the limit of 
their respective capacities. 

LESSON V 

UNIT OF INSTRUCTION II. — Book I 
RECTILINEAR FIGURES 

Lesson Type. — A Deductive Lesson 
Program or Time Schedule 

The Review 30 minutes 

The Assignment 5 minutes 

The Study of the Assignment 25 minutes 

The Review. — Subject Matter. Original exercises. 
Method. The figures for the exercises which were assigned 
in yesterday's lesson and which are to be reviewed to-day 



130 Supervised Study in Mathematics and Science 

should be placed upon the blackboard prior to the assembling 
of the class. By a survey of the A notebooks, handed in 
at the close of the period the day previous, the teacher will 
have noted those examples which have given trouble. This 
may be easily kept in mind by a simple method, that of 
noting on the back of the assignment sheet, or any piece of 
paper, the examples by number and beside the numbers the 
names of the pupils having difficulty with them. The exact 
trouble might also be indicated. These memoranda might 
read as follows : 

No. 1. O.K. 

No. 2. O.K. 

No. 3. John Jones (does not understand meaning of midpoint). 

No. 4. O.K. 

No. 5. Ben Ayers. 

No. 6. John Jones. 

Ethel Clare. 

Mollie Pond. 

Assuming, then, that all the pupils solved the first two 
examples correctly, and that one pupil had trouble with 
the third, begin with that one. When all have solved a 
certain exercise, there is no object in giving it any more at- 
tention. 

Ask John to rise and read the third example, which is as 
follows : 

Given 0, the midpoint of the line AB y CO ±AB, P any point in 
CO. Prove AAPO=AOPB. 

Then tell John to close his book and tell you the example in 
his own words. Until he can do this he is unable intelligently 
to consider its solution. Glancing at your notes, you will see 
that he seemed not to sense the meaning of the word " mid- 



Divisions of Plane Geometry 131 

point." After he has correctly stated the example, ask him 
what is given. Then ask him what he understands by the 
word " midpoint " and upon his failure to tell you, call on 
someone else to explain its meaning and significance in con- 
nection with this example. When John realizes that divides 
the line AB into two equal parts or segments, he may be able 
to proceed. If not, direct him by skillful questioning until he 
understands just wherein the key to the situation lies. It is 
probably in the fact that CO being perpendicular to AB makes 
equal angles at because they are right angles, and all right 
angles are equal. John is not told this, of course, but it is 
brought out either through questioning or is answered by some 
other pupil. Thus by guiding him to see significant facts 
concerning the example, which he failed to see by himself, 
he is being taught how to study and will feel within him- 
self a growing power which will stimulate him to greater 
efforts. 

Now that he has been guided through the solution, tell him 
to start over and go through it again. This may take longer 
than the teacher feels ought to be devoted to one pupil, but it 
should be of value to the rest of the class at the same time, 
and that a few are directed to find their difficulties and solve 
the problem that gave them trouble is worth much more 
than many exercises hastily done and nothing positive 
accomplished. It is not necessarily the number of exercises 
that one does which counts but the ability to do them by one's 
self. It is of more value to a class to conquer a few problems, 
even if done painstakingly and with toil, than it is to work 
many which require practically no effort. The teacher may 
feel well repaid if each day he can help a small number to 
make a definite advance in the mastery of the subject. 



132 Supervised Study in Mathematics and Science 

Repeat the above procedure with the fifth exercise, the next 
one to give trouble, always giving the problem or proposition 
to the pupil who experienced difficulty with it. Those who 
had no difficulty may at the same time feel that they also are 
advancing for they have the assurance that, through their 
additional work with the maximum assignments and their 
work as shown by the B notebooks, they are acquiring a real 
grasp of the subject. 

When the examples giving trouble have thus been disposed 
of, if time remains, a few similar examples from other texts 
may be given to those who successfully completed the mini- 
mum and average assignments. If there are some who have 
in addition completed part or all of the maximum assignment, 
they might be given one of special difficulty and either sent to 
the board or told to work at their seats. Excellent supple- 
mentary material may be found in Schultze and Sevenoak's 
Plane Geometry. 1 Special attention is called to the practical 
applications of geometry in the back of this book. 

The Assignment. — 1. Study of the notebooks. 

2. Explanation of the assignment. 

Directions for Study of the Returned Notebooks. Return 
the A notebooks to the class and explain that you have 
checked the exercises which have been done correctly with a 
" C " and that you have marked errors made in those done 
incorrectly, calling their attention to the thing that caused 
them trouble. Tell them to look these over first and to note 
your remarks. If they do not understand the remarks, remind 
them that they may call you to them and you will explain in 
more detail. If their work shows no errors, they may proceed 
at once with the advance assignment. 

1 The Macmillan Company. 



Divisions of Plane Geometry 133 

Explanation of the Nature of the New Assignment of 
Exercises. Call the attention of the class to their new assign- 
ment upon the board. Explain that the exercises are similar 
to those of to-day, but that some lack the figures. Call their 
attention also to the rules you gave them yesterday for study- 
ing originals and explain that these should be applied to the 
study of each item in the assignment. 

The Study of the Assignment. — I or Minimum Assignment. 
Exercises 94, 95, 96 (with figure), 97, 98, Schultze and Seven- 
oak's Plane Geometry, page 26. 

II or Average Assignment. Exercises 99, 100, 101, 102 
(same source). 

777 or Maximum Assignment. Exercises 108, 112, and 115 
(same source). 

The Silent Study. — As soon as all are at work on the new 
lesson, look over as many of the outside or B notebooks as 
possible, making needed criticisms as before. These errors 
may be called to the attention of the pupils at once, or at the 
close of the period. The amount of time that the teacher will 
be able to devote to this will depend on the amount of time he 
will be called upon to give to pupils with their new work. But 
since the pupils will resume the study of propositions to-mor- 
row, it is not necessary to review all the work at this time. 
The work may be collected to-morrow and checked as soon 
as convenient. Sometimes the B notebook may not be 
required and the work done outside of class simply handed 
in on loose paper. This may be found to be a more satisfac- 
tory way under some conditions, but the notebook that 
includes the work done in class is felt by the writer to be very 
important and the work so preserved is always available for 
purposes of comparison or review. 



134 Supervised Study in Mathematics and Science 

LESSON VI 

UNIT OF INSTRUCTION U. — Book I 
RECTILINEAR FIGURES 

Lesson Type. — Socialized Review Lesson 

Program or Time Schedule 

The Review 60 minutes 

The Review. — Subject Matter. All propositions of Book I. 

Method. Have the various theorems of Book I written 
upon small cards about two by three inches in size. Shuffle 
these and, after sending as many pupils to the board as you 
have panels, let each one draw a card and proceed to draw the 
figure and write out the demonstration. After the pupils 
have worked awhile, send the one at the extreme left to his 
seat and request the others to move one panel to the left, and 
to go on with the demonstration where the other left off. 
A new pupil is sent to take the place of the one seated and to 
work the proposition at the extreme right. This process may 
be repeated at intervals of a few minutes or done only once, 
depending on the judgment of the teacher. When an exercise 
is completed, send another pupil with a new card to the board 
to start a new proposition. 

This method of reviewing a number of propositions will 
entail considerable work on the part of the teacher, but by care- 
ful planning it may be very readily administered. Pupils at 
their seats should in the meantime have been given the same 
kind of work to do on paper, which may or may not be inter- 
changed from time to time. 

Some Advantages of This Kind of Reviews Are: 

1. All are working. 

2. Each pupil is reviewed on a number of propositions. 



Divisions of Plane Geometry 135 

3. The pupil's work is under the scrutiny of teacher and 
pupil. 

4. It emphasizes the necessity for neatness, thoroughness, 
and legibility. 

5. It develops judgment on the part of the pupil who takes 
up the unfinished work as to the correctness of that already 
done. 

6. It socializes the work of all the class ; each one is de- 
pendent not only on what he has done but on what the pre- 
ceding pupil has done. Many modifications will occur to 
different teachers. For instance, it may be better to have 
the pupils finish their demonstrations and then assign others 
to look them over and report ; if any are incorrect, have these 
erased and reworked by the pupil, or he may be directed to 
make the corrections himself. 

Again, two pupils may be assigned to each proposition, one 
to write down the steps and the other to write down the cor- 
responding reasons in the second column. These two may 
change places on a given signal. 

LESSON VII 

AN EXHIBITION OR RED LETTER DAY LESSON 

Object. The object of a lesson of this nature is threefold : 

1. To bring out the practical nature of geometry. 

2. To arouse in the class the desire to do superior work. 

3. To further interest in the subject. 

Place. This type of lesson, although not to be overempha- 
sized, may with profit be given at the close of the work on 
each book. 

Preparation. Like all stated programs, the lesson should be 



136 Supervised Study in Mathematics and Science 

planned a week or more in advance, the different assignments 
carefully made, and the purpose well explained, and it should 
be carried out according to plan. 
Method. The program may consist of three parts : 

1. Work to be put on display. 

2. Papers and oral proofs. 

3. An exposition of some particular phase by the teacher, 
principal, or superintendent. 

Explain early in the study of the subject that you are going 
to have an exhibition at the completion of this book of the 
best work that has been handed in during the period covered. 
This may consist of the best notebooks, best construction 
problems, best designs, etc. Tell the pupils that from time to 
time you will select the best drawings for special attention, 
such as having them traced in India ink on high-grade paper. 
Encourage the pupils to do their work from day to day with 
this exhibition in mind. 

Assign someone to prepare a paper on some geometrician's 
life or some interesting phase in the history of the subject, or 
some equally interesting topic. Naturally the display work 
will be done by the best pupils but encourage all to do their 
best to get on the program, and emphasize this as being an 
honor. 

The third part of the program will be taken care of by the 
teacher. 

Program. Keep the day of the special program before the 
class by skillful advertising, so that all will be on the qui vive 
for its approach. The teacher has had it in mind for some 
time and will have formed a clear outline of just what he 
wants done. With careful planning, the following program 
will be found of interest and at the same time instructive : 



Divisions of Plane Geometry 137 

Part I. Display 

1. Two or three of the best notebooks. 

2. A few of the best construction problems, drawn on high-grade 
drawing paper and traced in India ink. 

3. A few geometric designs, 1 done by maximum pupils and 
traced in colored inks. 

Part II. Papers and Oral Proofs 

4. Paper on Thales, who enunciated the first proposition, as 
well as many others. 

5. A recreation problem 2 proving that every triangle is isosceles, 
by the pupil with the highest grade. 

6. Model demonstration of some proposition in Book I, by some 
pupil, who will give it orally and without a figure. 

7. Paper on the use of the protractor for measuring angles. 

Part III. An Exposition 

8. Talk on the application of the truths learned in Book I in 
the problems of life, as surveying, architecture, and designing. 

1 Excellent material may be found in Sykes' "Source Book of Problems in 
Geometry "; Allyn and Bacon. 

2 Wentworth and Smith, "Plane Geometry " ; Ginn and Co. 



THIRD SECTION 

ADVANCED MATHEMATICS 



CHAPTER FOUR 

SPECIAL METHODS OF SUPERVISED STUDY IN HIGHER 
MATHEMATICS 

Intermediate and Advanced Algebra. — The subjects con- 
sidered in this chapter are usually offered to pupils in the last 
two years of their high school program. These pupils, through 
the study of elementary algebra and plane geometry, which 
it is assumed have been taught on the supervised study plan, 
should have mastered by this time the technic of how to 
study, and should be able to handle this advanced work with- 
out the detailed directions heretofore necessary. Consequently, 
the inspirational preview and how to study lessons will only 
rarely be required. The general plan outlined in the preced- 
ing lessons may be followed if preferred, but, since the ideal 
method would be for the pupils to advance individually and 
as rapidly as they are capable of doing, the suggested 
lessons herewith given will be based on this plan. With this 
in mind, and recognizing that pupils electing these courses will 
be those having more or less marked mathematical ability, 
we shall naturally expect that some will be able to cover part 
if not all the subject matter of intermediate and advanced 
algebra during the semester. 

In the same manner and with the same expectation we shall 
treat solid geometry and plane trigonometry as a unit. The 
time spent on each course being assumed to be twenty weeks, 
the scheme will probably work out about as follows : If at 
the end of the first ten weeks, some have covered at least three 

141 



142 Supervised Study in Mathematics and Science 

fourths of the work in intermediate algebra or the same amount 
in solid geometry, they will be expected to complete both 
courses by the end of the term. Those who have not advanced 
so far will finish the original course and spend the remaining 
time in review. This plan provides, therefore, for each pupil 
to proceed as rapidly as he is capable of doing and yet with 
no detriment to himself if he finds he is unable to complete 
both courses. The work will thus resolve itself largely 
into individual instruction, with a minimum of class demon- 
stration. 

Method. Since a large part of intermediate algebra is a 
review of elementary algebra, and since some pupils will need 
but little review and drill while others will require more, after 
a preliminary explanation of the plan for individual advance- 
ment, a test or series of tests covering the work of the elemen- 
tary course should be given to enable the teacher to judge just 
what preliminary or review study is necessary. This will 
naturally vary with the individual pupils and therefore resolve 
itself at once into personal direction. Those showing a per- 
fect knowledge of elementary algebra will immediately be set 
at work on the first advanced topic; those displaying a 
mastery of all except a particular topic (for example, com- 
pleting the square) should be assigned work on this unit of 
instruction, and so on. Some may have forgotten many of 
the details of the preceding course and, if there be enough of 
such pupils, they may easily be separated from the others and 
given some class instruction. In other words, each pupil 
should have his knowledge of elementary algebra analyzed, 
and he should be set to work upon the things he has forgotten 
or knows imperfectly. By intensive work with the individ- 
ual pupils, such deficiencies soon may be overcome. 



Study in Higher Mathematics 



!43 



Figure V 



Keeping a Card Index of the PupiVs Advancement. The 
teacher should keep a record of each pupil, preferably on the 
card index plan, noting thereon what each pupil is doing, when 
he has mastered each unit, and other pertinent data. If the 
units of instruction of the courses in algebra have been eval- 



144 Supervised Study in Mathematics and Science 

uated in some such manner as was done prefatory to the 
illustrative lessons in elementary algebra in Section I, this 
record may be kept in a very efficient way. The six column 
charging card, used by librarians, has been found very satis- 
factory in this connection. (See Figure V.) In the first 
column is checked the number of the unit of instruction, in 
the second column is noted the date it is begun, and in the next 
column the date it has been completed to the satisfaction of the 
teacher. The next three columns may be utilized for memo- 
randa concerning the difficulties experienced, test marks, etc. 
One of these cards for each pupil is easily administered and 
will effectually enable the teacher to " keep tab " on each pupil. 

When the class assembles, each pupil will immediately 
begin work on his individual task. He will request the teacher 
to render assistance if it is needed. The teacher, on the other 
hand, when not so employed, will pass from pupil to pupil, 
noting progress, checking results, and outlining advanced work. 
The schoolroom thus becomes a busy workshop, with every- 
one engaged on his particular problems, and no time is wasted. 
The pupils will respond eagerly to this method, because they 
will feel that they must proceed just as rapidly as they can 
with thoroughness. 

The mastery of each topic by the pupils may be effectually 
checked by the time-honored test, but it should be given by 
units and as soon as the pupil considers himself proficient 
enough to take it. The efficiency of the correspondence 
school is thus realized with the additional benefits derived from 
personal contact with the teacher and the momentum and 
inspiration of the classroom and its environment. Whatever 
additional work it may cause the teacher is counterbalanced 
by the high rate of efficiency and the elimination of ordinary 



Study in Higher Mathematics 145 

routine of the recitation. This will offset any possible hardship 
of added labor. It is not work that wears out the teacher, but 
worry; if the results are satisfactory, very few teachers will 
be found to complain of the effort demanded. 

Solid Geometry and Plane Trigonometry. — It is again 
assumed that pupils electing these courses have had those 
subjects already treated. Through the process of the sur- 
vival of the fittest, the class will undoubtedly be composed of 
those pupils with natural mathematical ability. The plan, as 
already outlined, will therefore prevail with some modifications, 
due to the nature of these subjects. 

With all superfluous work eliminated, pupils of natural 
ability ought to be able to master both subjects in twenty 
weeks' time. The ability to comprehend and master the text- 
book demonstrations having been acquired in plane geometry, 
the study of these in solid geometry should be very easy. A 
few class explanations of the use of the third dimension might 
be necessary at first, until the pupils realize and are able to 
visualize the figures. The method of proof is practically 
identical with that of the demonstrations in plane geometry 
and the number employed is much less ; and the amount of 
work in originals is greatly reduced. Since not all pupils are 
likely to complete both solid geometry and trigonometry, the 
work in the latter may well be treated individually as each 
pupil reaches it. 

Method. As soon as a pupil considers himself master of a 
demonstration, he may call the teacher to his side and either 
prove the proposition orally or on paper as the teacher elects. 
For the work on circles, a large spherical blackboard should 
be used by the pupils and his figures thereon should be ex- 
plained in detail. The use of stereoscopic views for showing 



146 Supervised Study in Mathematics and Science 

the more complicated figures will do much to enable the 
pupil to grasp their construction. These may be studied by 
the pupil whenever there is need to encourage him to work 
them out for himself. When called on for aid, the teacher 
should as far as possible confine his help to directing questions, 
which will enable the pupil to discover the answer to his own 
inquiry. 

Summary. — The success of the above plan for the advance- 
ment of each pupil as rapidly as he is able to master the sub- 
ject rests on two essentials: (1) a thorough mastery of 
the technic of how to study, developed through a sys- 
tem of supervised study from the early grades, and (2) a 
teacher, thoroughly acquainted with this system and with an 
intelligent grasp of his subject, who is sympathetic and alert 
to the possibilities of the effective administration of this 
system. With such qualifications of the teacher and normal 
intelligence and application on the part of pupil, the results 
will be highly successful, because they are the logical and 
inevitable culmination of the system of supervised study. 



PART TWO 
SCIENCE 



CHAPTER FIVE 

THE MANAGEMENT OF THE SUPERVISED STUDY PERIOD IN 

SCIENCE 

Supervised study in its last analysis is essentially the method 
of the laboratory, and therefore science should and does lend 
itself ideally to its application. The study of a textbook is 
more or less subordinated to the study of actual materials, 
individual experimentation supplanting to a marked degree 
the assumption of certain facts and conclusions because 
stated as such in some book. In other words, the pupil is 
taught to verify what the author has said. 

The study of science, largely perhaps because of its peculiar 
method of approach, has always been more or less popular 
with young people. Growing boys and girls are enabled 
through the actual handling of and experimentation with real 
things to give expression to their very active desire to do things, 
and an interest is aroused which can never be imparted through 
the mere reading of the printed page. It is for this very rea- 
son that manual training, domestic art, and allied subjects are 
especially attractive. The adolescent youth yearns to deal 
with " objective realities " * and, when they assume in 
addition some practical aspect, their study seems peculiarly 
valuable. 

1 Lloyd and Bigelow, "The Teaching of Biology"; Longmans, Green and 
Co., 1914. 

149 



150 Supervised Study in Mathematics and Science 

Laboratory work, however, needs to be carefully supervised 
if the student is to acquire real benefits from it, as aimless 
tinkering with apparatus and equipment may become simply 
a waste of time. It is for this reason that the supervised 
study period in science is particularly well suited to show the 
benefits of the system. Pupils must be directed in their 
scientific study even more carefully than in the case of mere 
book study and with correspondingly larger results. They 
must be taught how to test and how to properly draw con- 
clusions ; how to check their personal observation with that 
of others ; how to use various kinds of supplementary material 
and how to judge it ; how to interpret printed directions and 
to form unprejudiced opinions ; in fine, how to study. 

Much that has been written in Chapter One is equally 
applicable in connection with the work in science. 

The Time Schedule. — The length of the period being 
assumed to be sixty minutes, the division of the time for the 
regular recitation periods may be made as follows : 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 

When laboratory work is desired, however, the entire sixty 
minutes may well be devoted to the experimental work and its 
recording. If five or ten minutes be taken at the opening of 
the hour, however, to explain what the object of the experiment 
is, it will be time well spent. And, again, much will be gained 
if a few minutes at the close of the period be used to clinch the 
essential results of the laboratory work. These variations 
will be more fully explained in connection with the illustrative 
lessons. 



Study Period in Science 151 

The Assignment Sheet. — The use of the assignment sheet, 
as explained in the foregoing chapter, is strongly recommended 
for the science teacher in order that he may have a clear out- 
line of just what he expects to accomplish within the hour. 

The threefold assignments are also urged if the highest 
efficiency of the class is to be developed. While there is 
certain work that must of necessity be demanded of all, there 
should always be some arrangement made by which those who 
are particularly gifted may be led to do advanced work and 
thus proceed at a maximum rate and so become more proficient 
in their knowledge of the subject. This may take the line of 
extra study, supplementary reading, use of special apparatus, 
making charts, classifying museum materials, attending to 
equipment, making of simple apparatus, etc. Obviously the 
teacher who is satisfied with a minimum or average amount 
of work being done alike by all is failing to realize that the 
schools should develop each pupil according to his individual 
ability. This possibility of leading all pupils to do their best, 
of inspiring them to strive always for maximum results, is in- 
deed one of the strongest arguments in favor of supervised 
study in science as well as in all departments. There is no 
other method of conducting classes which so nearly approaches 
the ideal of scientifically educating all the children to the limit 
of the individual capacity of each. 

Mimeographed Sheets. — The use of laboratory manuals 
is not recommended since they naturally contain many things 
that the live teacher will find it necessary to change, on account 
of the apparatus at hand or because he will find that many 
things he wishes to incorporate are not mentioned. Mimeo- 
graphed sheets, which allow the individuality of the teacher to 
be displayed, are much to be preferred. These may be 



152 Supervised Study in Mathematics and Science 

changed from time to time as the teacher develops new ideas. 
In lieu of these, the blackboard may be used, but less advan- 
tageously. 

Too much should not be stated on these sheets. One of 
the gravest faults with laboratory manuals is their complete- 
ness ; too often they leave nothing for the pupil to do but to 
" press the button." If the pupil is really to grow from day to 
day in his ability to perform experiments and draw correct 
conclusions, much must be left to his own ingenuity and re- 
sourcefulness. In cases where the pupil seems to have this 
ability undeveloped, the necessary explanations may better be 
made to him individually ; thus it becomes possible for each 
pupil to rely to a large extent on his own endeavors. In other 
words, the directions should be mainly suggestive, and details 
should be supplied by the teacher or pupil as the case 
requires. 

The pupils should also be encouraged to construct as much of 
their apparatus as they are able to make. Every well-equipped 
laboratory should have a workbench, fully supplied with 
ordinary tools, materials, etc., so that the pupil may have an 
opportunity to develop whatever ingenuity he possesses. 

The teacher should also be careful not to do too much him- 
self. His ingenuity in constructing apparatus and performing 
experiments may best be displayed by his ability to develop 
ingenuity and performance in his pupils. The pupil should 
be led to draw the proper conclusions and not told what con- 
clusions he should draw. The latter is rather dogmatic teach- 
ing but the former may well be called superb leadership. 



FOURTH SECTION 
BIOLOGY 



CHAPTER SIX 

DIVISIONS OF BIOLOGY 

In evaluating a course in biology, since, from the very nature 
of the vast amount of available material, it is impossible to 
cover all the phases, it is best to select certain topics in botany, 
zoology, and physiology which illustrate the fundamental 
principles of life. This method at the same time will give the 
boys and girls some important first-hand knowledge of plant 
and animal life, with special emphasis on the economic and 
hygienic sides. Since authorities will naturally differ on the 
topics that should be included, the divisions as here made may 
be open to criticism, but for the sake of concreteness those 
outlined by the State Department of Education of the State of 
New York are followed, with a few modifications. 

Units of Instruction 

I. Introductory topics. 

Plant Biology 

II. Seeds and seedlings. 

III. The cellular structure of living plants. 

IV. Roots. 
V. Stems. 

VI. Leaves. 

VII. Flowers and fruits. 
VIII. Forests and forest products. 

i55 



156 Supervised Study in Mathematics and Science 

Animal Biology 

IX. Insects. 

X. Crustaceans. 

XI. Fishes. 

XII. The frog. 

XIII. Birds. 

XIV. Mammals. 

Human Biology 

XV. Foods, stimulants, narcotics. 

XVI. Bones and muscles. 

XVII. Organs of digestion and their functions. 

XVIII. Blood and circulation. 

XIX. Respiration. 

XX. Excretion. 

XXI. Bacteria and sanitation. 

XXII. Additional topics, including the nervous system, 
special senses, first aid. 

Below is given a suggestive time table for the year's work in 
biology. This will of course vary with local conditions but 
is given merely for its possible suggestive value : 

First twelve weeks Plant biology 

Next twelve weeks Animal biology 

Next twelve weeks Human physiology 

Last four weeks Review 

LESSON I 

THE INSPIRATIONAL PREVIEW 

Purpose. — The purpose of such a lesson is to arouse the 
interests of the child in the study of biology. There is no sub- 
ject in the whole program of studies which may have as many 
practical applications to the welfare of the pupil as does the 



Divisions of Biology 157 

study of biology. But arousing the pupils to an appreciation 
of these values and incidentally to its many pleasures and 
interesting phases necessitates that the teacher make some 
effort to predispose the child's mind to receive its valuable 
lessons. This first meeting with the class presents this op- 
portunity. 

Method. Naturally, the method of conducting an inspira- 
tional lesson will vary with the teacher, since it will reflect his 
personal ingenuity, but for the purpose of possible suggestion, 
the following scheme, which has been successfully used by the 
author, is presented herewith. 

Previous to the assembling of the class for the first time, 
place a small quantity of grass and some grasshoppers in a 
fine-screened cage. As soon as the class is assembled, call 
attention to the cage and state that it contains illustrations, 
in a way, of the purpose of the study of biology. Ex- 
plain that biology is the study of life, animal and vegetable, and 
its practical applications to the existence of man. Man is not 
only dependent on these two kingdoms of nature for his suste- 
nance, but his raiment, his health, his vocations, and his avoca- 
tions are largely biologic in their nature. Draw attention to 
the grasses and plants in the cage and explain that the vege- 
table kingdom, of which they are examples, furnishes us with 
food, directly or indirectly, clothing, homes in which to live, 
and determines many of our most important occupations ; that 
some plants cause sickness and others offer the means of pre- 
venting or curing our ills, others give us pleasures, etc. 
Explain that in order to know just how these things come about, 
it behooves us to know how the plant grows, how it lives, and 
how it reacts upon our lives. Tell the pupils that there is 
within this cage also a representative of the animal kingdom 



158 Supervised Study in Mathematics and Science 

— a large group of living things, on which in a large way we 
are likewise dependent for food, raiment, transportation, 
good health, and also many of the pleasures of lif e. Our work 
in biology will lead us into a systematic study of animal life 
and its part in the scheme of living. 

Now have someone come forward to open the cage and 
find a grasshopper within. The difficulty of finding him in 
his camouflaged natural habitat will give the teacher an 
opportunity to speak of the insect's environment, his protective 
coloration or adaptation, his dependence upon the plant king- 
dom, and, therefore, the close connection between plants and 
animals. 

Place the grasshopper upon the floor and let him jump. 
Measure the length of the jump and compare it with his size. 
Ask some pupil how high he can jump and make a like com- 
parison with his height. The wonderful adaptability of the 
hind legs of the grasshopper for this activity will appeal to the 
pupils as extraordinary, as it indeed is. Mention the fact 
that all the various classes of animals we shall study have like 
wonderful adaptations for their modes of lif e and also that they 
are beautifully made for the part they are to play in the scheme 
of nature. 

Now after a few words briefly explaining the various 
functions of living matter, write upon the board the following 
captions which may be said to form the background of our 
work: 

Classification. 

Habitat. 

Structure. 

Life history. 

Adaptations. 

Functions. 



Divisions of Biology 159 

If now the teacher will read some interesting incident 
showing the wonderful intelligence of some insect, as the ant 
from Romanes' " Animal Intelligence/' x or the fly from 
Fabre's " The Life of the Fly/' 2 or the wasp from Morey's 
" Wasps and Their Ways/' 2 he will arouse an intense 
desire to know more about these wonderful little creatures, 
and the year's work in biology will begin auspiciously and in- 
terest can be easily maintained. 

Then briefly tell the story of the Panama Canal and its 
gigantic failure until General Goethals attacked the biologic 
problems and made the region safe for man to inhabit, chang- 
ing failure to success. Tell about the work of Dr. Koch and 
his discovery of the cause of tuberculosis, which has resulted 
in its subsequent treatment ; the wonderful work of Luther 
Burbank in producing new varieties of plants ; the investiga- 
tions of Dr. Wiley in regard to pure food ; the work of the 
Federal Bureau of Entomology and its aid to the farmer. 
Explain that in every case it was the application of biologic 
facts that has made life more enjoyable, more healthful, and 
more successful. 

With such a bird's-eye view of the contributions of biology, 
the pupils will not fail to realize that here is a subject full of 
the practical and the interesting ; and they will actually feel 
the desire to master this subject in so far as they are 
able. 

If time permits, a rapid survey of the course may be made. 
Explain that first we shall make a study of the elements that 
enter into all lif e, both animal and plant ; then we shall study 
plants from the seedling to the matured plant ; animals from 
the simple one-celled species, which can only be seen under the 
1 D. Appleton and Co. 2 Dodd, Mead and Co. 



160 Supervised Study in Mathematics and Science 

microscope, up through the crustaceans, the insects, the fishes, 
birds, and mammals ; and finally we shall make a study of 
man, the most wonderful and highest creation of all. 

The pupils should be informed that they will be expected to 
bring specimens to class, from time to time, to add to the 
school's collection of natural history materials. Tell them 
to be on the lookout for articles relating to biology, such as are 
appearing constantly in newspapers, magazines, and books. 
Tell them that in addition to the study of their textbook we 
shall do more or less experimentation and investigation of 
actual specimens in the laboratory and in their natural envi- 
ronment. 

The teacher must be careful not to make this preview too 
technical or too stilted. It is very important to make the 
pupils forget that they are about to become formal students 
of a new subject which may contain a good deal of dry, uninter- 
esting drill on facts. Young people at this age are naturally 
more or less inquisitive and something which will make keener 
this instinct for the novel or the unknown will be worth while 
if at the same time it arouses within them a sincere desire to 
learn. 

The above suggested plan may be objected to on the ground 
that it smacks too much of the idea of nature study and is too 
childish. Are we not dealing with young children of immature 
minds and is not biology in fact nature study with the stress 
on its applications to life? In the best sense biology is or 
should be the study of natural history with the underlying 
thought of the continuity of life processes and functions, and 
its applications to the whole scheme of nature. Both experi- 
ence and pedagogical considerations seem to justify this view- 
point. 



Divisions of Biology 161 

LESSON II 

UNIT OF INSTRUCTION I. — INTRODUCTORY TOPICS 

Lesson Type. — A Lesson on How to Study 

Program or Time Schedule 

The Assignment 10 minutes 

The Study of the Assignment 50 minutes 

Purpose. — The purpose of this lesson is to introduce the 
pupil to the study of biology through some preliminary experi- 
ments with chemical elements and compounds. The textbook 
is at once to be supplemented with the actual handling and 
examination of biological materials, thus illustrating the 
laboratory method to be followed more or less throughout the 
course. It is desirable to make clear at the outset that the 
textbook is in the nature of a guide, to explain and organize our 
study. Just as guidebooks in travel may not only serve as a 
prospectus of what we may expect to find and see, but also as 
a means of interpreting those things which we see and wish to 
know more about, so our textbook in biology will guide us in 
our work to discover certain things and will supplement what 
we see with data concerning which we would otherwise remain 
in ignorance. 

The Review. — This will usually take the form of clinching 
firmly in the mind of the child the lessons learned previously, 
and the process should be pointed, short, and clarifying. As 
this is the first lesson besides the inspirational preview, the re- 
view may be dispensed with to-day. 

The Assignment. — In all laboratory work the assignment 
may well take but a few minutes and will be used to explain 
the nature of the new work. In the present case, it will suffice 



1 62 Supervised Study in Mathematics and Science 

to explain that we are going to spend the period examining 
some material as to its physical characteristics and as to what 
it will do under certain manipulations. 

Notebooks. Explain that the use of notebooks, or some 
means of recording our investigations, is necessary in all 
laboratory work in order that we may set down scientifically 
the facts we learn and that we may have them for future 
verification and review. Data so recorded must be accurate 
and as clear as we know how to make them. Explain that 
some system should be followed which will help to make 
the data clear. Each of the following heads should be 
employed : 

i. Date. 

2. Object. 

3. Material. 

4. Manipulations. 

5. Results. 

All pupils should be supplied with some form of notebook, 
the kind necessarily varying with the individual taste of the 
teacher. The end-open loose-leaf notebook, size about 4X6 
inches, is recommended. This makes a compact little book 
which may be carried easily in the pocket or bag on field 
excursions. State that these books will be collected from 
time to time and examined by the teacher, and emphasize the 
importance of keeping them as neat as possible and of 
always having them ready for use. 

The Study of the Assignment. — There should be placed 
upon the desk or table of each pupil all the materials which are 
to be studied. A little careful planning will obviate wasting 
of time during the period. In the present instance, each pupil 



Divisions of Biology 163 

should be supplied with small pieces of carbon (charcoal), iron, 
sulphur, and phosphorus (in water). 

Procedure. Tell the pupils to open their texts and read the 
paragraph on elements and compounds. As soon as they have 
had time to read it through understandingly, have them close 
their books and question them on what they have read. The 
teacher may well at this point expand somewhat the state- 
ments of the author, being careful not to confuse chemical with 
physical compounds. 

Then tell the pupils to examine the piece of charcoal with- 
out referring to the book, and after a few minutes question 
them as to its characteristics. Try to get them to tell what 
they have found out about the charcoal without your asking 
too many questions. Different pupils may be called upon to 
add something new to the peculiarities or characteristics al- 
ready given, until all the important ones have been mentioned. 
If this method fails to elicit all the desired information, it may 
become necessary to ask definite questions, such as, What is its 
taste? 

Repeat the process with the piece of iron, and then have the 
pupils compare the two elements for similarities and differ- 
ences. Insist on the results being stated in sentence form 
with due regard to the English used. Ask them to tell in what 
forms each of the two may be found and draw out therefrom, 
if possible, the significance of these characteristics. For 
instance, they have discovered that charcoal is soft and leaves 
a mark when drawn across paper. This quality has given 
rise to charcoal pencils used by artists. A similar character- 
istic of graphite has been utilized in the making of so-called 
lead pencils. Iron is found to be tough ; how has this char- 
acteristic been utilized in making iron stoves, rails, etc. ? 



164 Supervised Study in Mathematics and Science 

After they have done this, let them open their books and 
read over what the author has to say concerning these two 
elements. Someone might take another text in biology and 
read aloud to the class what the author has to say concerning 
this subject. Step to the board and write a list of things made 
from each element, items being suggested by the pupils them- 
selves. It is extremely important to have the pupils feel that 
they are furnishing the data and that the teacher is merely 
leading the way and correcting any wrong deductions. 

Recording the Experiments. The next step is to record in 
the notebooks the results of this exercise. Have the pupils 
open their notebooks and, following the order noted above, 
make their records. The one on charcoal should appear 
something like this : 

Plan of Notebook 

Date: September 4, 19 2-. 

Object: To find the characteristics of carbon. 

Materials: Piece of charcoal, a dish of water, a match, a knife, 
paper. 

Experiment: I took the charcoal in my hand, noted its weight, 
color, feel, taste, odor, and texture. I put it in some water to see 
whether it would dissolve, I tried to make it burn, I tried to cut 
it with a knife, and I rubbed it upon some paper. 

Results: I found that it was light in weight, black in color, felt 
rough, had no taste or odor, and looked to be rather porous. It 
will not dissolve in water, burns with a glow, and makes a mark on 
paper. 

Some other forms of carbon are : graphite (used in lead pencils) ; 
diamonds. 

Carbon results from burning wood, as the match, and is some- 
times found free in nature. 



Divisions of Biology 165 

Next, the pupil should take up the study of sulphur in a 
similar way. The work on phosphorus will need to be guided 
more definitely by the teacher, due to its peculiar character- 
istics. After the four elements have thus been studied and 
written up in the notebooks, if time permits, the pupils should 
study the paragraph on oxygen and the air, after which the 
same order of study may be followed. 

It is very important to make haste slowly in this new work. 
The pupils are immature and cannot grasp too many new 
principles at one time. Each phase should be covered thor- 
oughly, with all the variations that it is possible to make. The 
actual material in the textbook should be supplemented by 
other books, by the pupils' experience and knowledge and, if 
necessary, by the teacher. The illustrations and applications 
should be as varied and comprehensive as possible, for this is 
the way by which the pupils will learn to study. 

LESSON III 
UNIT OF INSTRUCTION I. — INTRODUCTORY TOPICS 

Lesson Type. — An Inductive Lesson 

Program or Time Schedule 

The Review 20 minutes 

The Assignment 15 minutes 

The Study of the Assignment 25 minutes 

The Review. — The work to date has been on the prelimi- 
nary experiments. Since we are now to start on a new topic or 
unit of recitation, it may be best to spend a little longer time 
than is given usually to review, in clinching the lessons already 
studied. This may take the manner of a short, snappy quiz. 



1 66 Supervised Study in Mathematics and Science 

In addition, we might make a formal testing of some phase of 
the work previously studied. These twenty minute reviews 
occurring every four or five days, partly oral and partly written, 
are much better than formal written examinations of length 
and given at longer intervals. These reviews test the knowl- 
edge of the class on many of the details; and through the 
short written test, they cover with definiteness some particular 
feature of the work and make it possible for the teacher to know 
just how completely the pupils have grasped important points. 
The question or questions — one is usually sufficient — should 
be broad and should be based upon some definite or underlying 
principle. For example, the class to-day, after the oral quiz, 
might be asked to write a paragraph on the importance of 
reducing all compounds to their elements. 

Allow ten minutes for this paragraph and then collect the 
papers. These should be carefully examined and errors noted, 
and two or three of the best answers read aloud to the class 
the next day. If none is satisfactory, the reasons for this 
fact should be summarized and possibly a model answer writ- 
ten upon the board. Pupils will thus learn how to give sat- 
isfactory written answers. Since written examinations as a 
means of judging proficiency in a subject are bound to be with 
us for some time to come, this is an effective method of pre- 
paring the pupils for them. By practice and through criticism 
each pupil will learn the kind of complete and definite state- 
ments that will be expected from him in these written tests. 
As already noted, fact questions need not be emphasized in 
written tests since the oral quiz is just as satisfactory and is 
a time saver. 

The Assignment. — Explain that many of the lessons in 
biology will be in the nature of problems. To-day's problem 



Divisions of Biology 167 

is: Are two plants or parts ever alike? Our study will be 
based upon illustrative material and the textbook. 

Tell the pupils to make a list of articles, found in nature, 
which are similar in form. After a few minutes ask someone 
to read the name of the first article on his list. It might be 
" man." Ask whether two men are ever exactly alike. The 
pupils will readily answer that they are not. Repeat with 
others on the lists, as dog, potato, stone, apple, etc. The 
pupils will agree that all seem to differ in some particular from 
each other. Ask whether things in the artificial world so 
differ from one another. They are more likely to be seemingly 
alike but, even in the case of the textbooks in use in class, each 
book may differ somewhat from its neighbor. Some may have 
pages uncut, some may have defects in binding, some may 
have pages poorly printed, etc. ; yet they differ, if at all, in 
degrees of perfection. Then direct the pupils to study the 
maple leaves which will have been placed in advance upon 
their desks. 

The Study of the Assignment. — Each pupil having been 
supplied with a number of leaves, ask them to find two alike 
if they can. They may exchange with each other if they 
wish. If some pupil thinks he has found two exactly alike, it 
may be necessary for the teacher to point out some difference 
in veining, hairiness, markings, color, etc. It would be an 
excellent thing if a quantity of leaves of a different kind could 
also be passed around for examination. The greater the 
variety of specimens for examination the more scientific and 
impressive will be the lesson learned. A quantity of twigs 
from the same tree should also be secured for a like investiga- 
tion. After ample time has been allowed for all to satisfy 
themselves that no two specimens are entirely alike, have the 



1 68 Supervised Study in Mathematics and Science 

pupils open their textbooks and read what the author says 
concerning this problem (Bailey and Coleman's First Course 
in Biology, pp. 1-3). l Draw the attention of the class to 
the cut which also justifies the same conclusion, noting, how- 
ever, that a study of the specimen itself is to be preferred 
always to a picture. 

Tell them to read carefully the last sentence and then ex- 
plain in simple manner what the author means. This word 
variation, then, is the key word of our lesson to-day. 

The pupil may never have appreciated so fully before the 
wonderful ways of nature as he will now when he comes to the 
realization that, of all the billions of maple leaves in the world 
to-day, as well as in all the years gone by, there are never two 
exactly alike. Here is an opportunity for the teacher to im- 
press the child with an appreciation of nature through scientific 
observation. 

The observations listed on page 3 of the text with regard to 
size may well be taken up next and each pupil told to answer 
these questions concerning one of the leaves he has before 
him. If time permits, it would also be beneficial for him to 
note these facts in his notebook. 

Before this is done, however, the result of the day's 
investigations should be noted and cast into a form of state- 
ment ; and it may with good effect be written upon the board 
with colored crayon. Ask someone to tell what conclusion he 
has drawn from the day's lesson. If the pupil has been alive to 
the problem under discussion and examination, he will say, 
" No two plants or parts are ever alike." 

(N.B. Note the title of the chapter referred to.) 
1 The Macmillan Company. 



Divisions of Biology 169 

LESSON IV 
UNIT OF INSTRUCTION I. — INTRODUCTORY TOPICS 

Lesson Type. — An Inductive Lesson 

Program or Time Schedule 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 

The Review. — On each of five panels of blackboard, have 
these directions : 

Write a list of points in which leaves may vary from each other. 

Write a list of various kinds of leaves. 

Make a list of flowers which do not at all resemble each other. 

How do people resemble each other ? 

How do people differ from each other? 

Assign a pupil to each of these sets of instructions, directing 
him to write the answers upon the board. Meanwhile, ask 
someone for the conclusion reached in the preceding day's 
lesson, and ask different pupils for proof of its accuracy. 

The lists having now been completed, read each exercise and 
the answers given. Remarks may be necessary to amplify or 
correct mistakes ; have other pupils supply this material where 
possible. 

The Assignment. — Explain that the new lesson is to be 
based upon a study of a continual warfare that is being waged 
in the plant and animal world ; namely, that of a struggle 
for life. Remark, while one may go to the mountains for the 
summer and meditate upon the tranquillity of nature, that in 
fact a very serious strife is being waged by every tree and 
flower and living thing. Each is making a desperate struggle 



170 Supervised Study in Mathematics and Science 

for a chance to live, and, just as each is successful or un- 
successful, will the existence of a particular flower or bird be 
determined. 

Ask the pupils whether they can relate any experience 
in observed animal or plant life which would go to prove 
that this is so. Many examples in the animal world will 
come to their minds of one animal's living upon another. 
After a few illustrations have been given, ask whether 
anyone can give instances from the vegetable world. This 
may be a poser, but if you suggest weeds, the class will follow 
the clue. 

After this factor in plant life has been quite fully covered, 
explain that these plants and animals solve their problems by 
various means, such as the bird's escaping from the cat by 
rapid flight. This instance will set the pupils thinking, and 
the teacher will need the next few minutes to give them an 
opportunity to state similar cases of adaptation to con- 
ditions. 

State finally that these conditions, to which a plant or animal 
is obliged to adapt its habits of life or die, is called its environ- 
ment, and bring out the various conditions of environment, 
such as climate, food, habitat, etc. 

The Study of the Assignment. — The advance lesson will be 
the chapter on " The Struggle for Life." The essential fea- 
tures of this chapter have been already covered in our study, 
but without any reference to the book itself. Now through 
the study of the assignment or the textbook, the pupils are to 
organize this material into a clear, concrete increment of 
knowledge. Some words may need explanation; pupils 
should early feel that the dictionary is to be a constant help 
and should be referred to frequently. If any pupil finds some- 



Divisions of Biology 171 

thing in the lesson which he cannot understand, he should 
raise his hand and call the teacher to his aid. 

An added exercise should be the requirement to write out a 
list of at least ten cases of struggle for existence, ten adapta- 
tions to conditions, and ten varieties of environment. 

LESSON V 
UNIT OF INSTRUCTION II. — SEEDS AND SEEDLINGS 

Lesson Type. — A How to Study Lesson 

Program or Time Schedule 

The Review 10 minutes 

The Assignment 15 minutes 

The Study of the Assignment 35 minutes 

The Review. — Each pupil having been assigned some plant 
for special report to-day on the essential points of classification, 
the review may be spent on this work. Each pupil in turn 
may be called upon to name the plant assigned to him for study, 
and to classify as annual, pseud-annual, plur-annual, biennial, 
etc., and to state clearly his reasons for his answer. (A sugges- 
tive list is given in Bailey and Coleman's First Course in Biol- 
ogy, page 19. Y 

The Assignment. — Preparatory Work of the Teacher. 
Each pupil should be supplied with a specimen of the bean and 
a kernel of corn, which have been previously soaked in water. 
Each pupil should also have a bottle containing a solution of 
iodine. A large chart showing the essential parts of these seeds 
should also be in full view of the class. In lieu of a chart, the 
teacher might make large drawings upon the blackboard. 
1 The Macmillan Company. 



172 Supervised Study in Mathematics and Science 

The teacher should explain in the beginning that the lesson 
to-day concerns seeds, their essential parts and their functions. 
He should then explain how to open a seedling, point out the 
embryo, and indicate how the test for starch should be made. 
He should also explain that these seeds have been soaked in 
water in order to study them more easily and to show the 
effect water has on them. 

The class should then be instructed to open the books 
and to study the text carefully, verifying each statement from 
examination of the specimens. When they have opened their 
books, but before they start their work, it will be well to ex- 
plain the correct pronunciation of the names of the various 
parts of the seed. These are new words and it is best to make 
sure that the pupils learn from the first just how they are to 
be pronounced. The derivation of some of the words might 
well be given, as plumule from the French word plume mean- 
ing to ascend and hence given to that part of the seed which 
will rise or ascend. Explain that monocotyledon is formed 
from two Greek words, mono, meaning one, and cotyledon, 
meaning a cup-shaped hollow, and signifies a seed having one 
cotyledon. Note that monoplane is an airship having one 
wing or plane. Again, endosperm is formed from the Greek 
adverb meaning around, and sperm, which signifies the life-giving 
element, hence the word denotes that which is around or encloses 
the embryo or sperm. The pupils thus having learned the 
reasons for the new names will find them not simply unfamiliar 
and unmeaning terms but living words, and they will experience 
no particular difficulty in learning to use them correctly. 

All of the words which are likely to give trouble should be 
marked in the teacher's book and definitions or explanations 
of them prepared prior to the meeting of the class. It is this 



Divisions of Biology 173 

careful preparedness on the part of the teacher that will do 
more than anything else to dispel confusion and that will 
result in the interested and undivided attention of the as- 
sembled class. 

The Study of the Assignment. — The pupils will now begin 
the study of the new work. The first sentence presupposes 
the opening of the seed and the discovery of the embryo. 
This then should be done, the bean being used first. The 
next sentence, we shall assume, describes the three essential 
parts of this embryo. With the direction of the teacher and 
the aid of the large charts, these should be found in the speci- 
men and the proper name for each learned. If any pupil 
has difficulty in finding these parts in his specimen, he should 
call the teacher at once to his side. Let it be definitely under- 
stood that no one is to proceed until he has covered thoroughly 
each individual point of the text. 

After this has been done with the bean seed, tell the pupils 
to open up the kernel of corn and trace its embryo through a 
like investigation. The teacher should follow the actual work 
of the pupils throughout, keeping pace with them in their 
study and supplying any details necessary to make the work 
clear. Ninety per cent of the textual work in a science must 
be studied in this painstaking and critical manner, nothing 
being passed by until mastered and, if possible, verified from a 
study of the specimen or material itself. We simply cannot 
read scientific literature over rapidly, looking for the high 
spots, as is sometimes done in some subjects, but each state- 
ment must be closely and carefully studied. It is for this 
reason that supervised study in science becomes very im- 
portant, and why study without it often results in superficial 
knowledge or utter failure. 



174 Supervised Study in Mathematics and Science 

Not too much should be assigned for a lesson. Better a 
little well studied and principles well grounded than to attempt 
too much. An excellent method of varying the order 
suggested in the above time schedule would be to use the last 
five or ten minutes of the period for a review of the essential 
points studied in the new lesson, thus firmly clinching them. 
Step to the chart, for instance, and call on someone to name 
the different parts of the seed, someone else to tell the function 
of each, and someone else to spell the various names. 

Before the class is excused, some of the unopened soaked 
seeds should be planted in a suitable box and placed in a sunny 
place ; some similar seeds which have not been soaked should 
also be planted in another box and the two boxes labeled, one 
as A and the other as B. It is suggested that some 
pupils be assigned to this duty, preferably someone who has 
been a little more industrious than the others and who may 
have completed the assignment. This will serve to keep all 
busy and will give some recognition to the more rapid workers. 
Speed should never be tolerated in place of thoroughness, 
however. 

If there be time for any more work during the period, the 
pupils may be instructed to make a drawing of the opened 
seed in their notebooks, labeling each part. Under the title of 
Seed Germination, the pupils should also commence a note- 
book record of the planting of the soaked and unsoaked seeds. 
The record will be added to later. If the work has been care- 
fully planned in advance, all of the required details can be 
completed within the hour ; the exact amount, however, will 
necessarily vary with teachers and classes. Sufficient work 
should be planned by the teacher for an emergency, always 
keeping the time limit of the period in mind so that there may 



Divisions of Biology 175 

be no undue hurry and no possibility of leaving some task half 
done. As has already been emphasized, it is very desirable 
to have sufficient time before the close of the period for a rapid 
review and the clear affirmation in some sort of summary of 
the important points of the lesson. 

LESSON VI 
UNIT OF INSTRUCTION II. — SEEDS AND SEEDLINGS 

Lesson Type. — An Inductive Lesson 

Program or Time Schedule 

The Review 10 minutes 

The Assignment 25 minutes 

The Study of the Assignment 25 minutes 

The Review. — Ask some general review questions on the 
preceding lesson, such as : 

Name some dicotyledons, monocotyledons. 

Where is the food stored in each ? 

Of what use is the seed coat ? 

Are all seeds alike ? Are all bean seeds alike ? 

Why is it poor policy to plant old seeds ? 

What care should be taken of seeds before they are planted ? 

A few such questions, combined with a quick review of the 
parts of the seed from the chart or drawing, will suffice to 
cover the review. 

The Assignment. — The new lesson will be similar to the 
preceding one in that it will be a continuation of the study of 
the textbook, exemplified in every possible particular by the 
use of real specimens. There will be also the starting of 
several experiments which from their very nature must take 



176 Supervised Study in Mathematics and Science 

three or four days for their completion. These experiments 
will be along the line suggested by the paragraph on germina- 
tion in the new lesson. The author states that " when 
supplied with moisture, warmth, and oxygen (air), it (the 
embryo) grows." l We wish to verify these facts. We have 
already started an experiment to discover the value of moisture 
as a factor of rapid growth by our planting of soaked and un- 
soaked seeds. We shall now make another experiment. 
Tell the pupils that every time we make an experiment, it 
is necessary that we have a so-called check or control experi- 
ment, the one to show that certain things happen under certain 
conditions and the other to show that without these conditions 
they will not happen. 

The pupils should be told to open their notebooks and, 
under separate heads, note what is done to prove the three 
things stated in the above quotation. 

Experiment I: Do seeds need moisture? 

Procedure: Let each pupil plant a seed, bean or corn, in a plant 
dish. Designate two pupils to keep it watered. Tell all the pupils 
to consider this C in their notebooks, and so label the dish. In 
another dish, marked D, plant some of the same seeds, remarking 
that this dish will not be watered. 

Experiment II: Do seeds need warmth? 

Procedure: Have the pupils plant seeds in two dishes, both to be 
kept watered, one to be set in the sunlight where it is warm, the 
other set aside in some cool place. Label them E and F respectively. 

Experiment III : Do seeds need oxygen? 

1 Bailey and Coleman, "First Course in Biology " ; The Macmillan Company, 
1908. 



Divisions of Biology 177 

Procedure: Make the double planting of moistened seeds in 
dishes labeled G and H, one to be left uncovered and the other 
securely corked. 

As soon as the pupils have made the requisite records in 
their notebooks, pass on to the study of the assignment. 

The Study of the Assignment. — Study carefully the 
assigned pages in the text which we shall assume cover the 
germination of seeds. Although the seeds planted yesterday 
of course have not sprouted yet, the teacher should have some 
beans in the various stages of growth ready for study, having 
planted them at various intervals previous to this time. When 
those planted by the class have begun to come up, the class 
should use them for review and for their drawings. They will 
thus serve as a verification of the textbook and also as the 
models for the drawings. 

It will be seen that the teacher must be continually planning 
for the future : by having seeds at the right stages of germina- 
tion; by having plants at the right stages of development; 
by having plenty of illustrative material ready at the proper 
time ; by keeping note of the progress of various experiments ; 
and by taking care that all phases of the subject are studied 
in their right sequence. A great deal of this may be done by 
the pupils themselves, outside of school hours. The teacher will 
find that they will be glad to do this work. As far as possible 
pupils should be rotated in these preparations, so that all may 
feel a corresponding responsibility. The biology laboratory 
should be at all times full of growing plants and flowers ; there 
should be an aquarium, cages for living animals, herbariums, 
etc. The pupils should feel that their biology room is es- 
sentially a place of living things, and they should be encouraged 
to contribute as much material as they can from time to 



178 Supervised Study in Mathematics and Science 

time. The pupils should also be encouraged to make similar 
experiments at home and to compare their results with those 
obtained in the laboratory. Incidentally it might serve to 
arouse a cooperative interest in their school work on the part 
of their parents. 

There should also be plenty of cabinet room for permanent 
specimens. The pupils should be encouraged to add to this 
collection either temporarily or permanently. In this way a 
very adequate museum may be accumulated which may be 
used for demonstration purposes. The teacher should em- 
phasize at all times, however, the greater importance of living 
things since biology is essentially the study of life. The 
author has little use for biological specimens purchased from 
commercial houses; some specimens, which are foreign to 
the locality, must necessarily be purchased, such as star- 
fish, etc. These, however, have only a subordinate value; 
the pupils' environment should furnish the mass of illustrative 
material. If it is a rural community, the specimens should be 
those of the farm ; if an urban community, those of the locality 
are to be emphasized. 

More can be learned about birds from some pupil's bringing 
his pet canary to school or from a trip to the woods and study- 
ing the birds as they are in their natural habitat, than from 
cases filled with mounted specimens of birds, native to foreign 
localities. At best, all the latter can do is to serve as means 
of identification, and it is doubtful if a child will ever learn to 
know the birds except from the study of the live birds them- 
selves. Let us inculcate a love for wild life in its natural 
setting, even if the specimens be limited in number ; at least, 
let us attempt to make biology a study of what it is generically 
intended to be, that of fife and living things. 



Divisions of Biology 179 

LESSON VII 
UNIT OF INSTRUCTION II. — SEEDS AND SEEDLINGS 

Lesson Type. — A Socialized Lesson 
Program or Time Schedule 
At least 60 minutes 

Purpose. — The purpose of this lesson is to study nature in 
its own setting, to learn some of the lessons from nature's great 
laboratory, the out-of-doors. Field excursions are absolutely 
essential to the proper evaluation of biology, and the more of 
them the better ; but let them always have some specific 
object in view, some particular lesson to be learned. Inci- 
dentally many other lessons will be learned, but some out- 
standing purpose for each trip should govern the excursion. 

As soon as the class assembles, explain that you are going to 
take a field trip for the main purpose of studying the ways in 
which seeds are scattered or dispersed. Tell the pupils to 
have their eyes open for the observation of anything else which 
pertains to botany, especially any illustrations that would 
seem to verify the lessons already learned in class. But the 
main object of the expedition is to study seed dispersal. All 
the pupils should have their notebooks handy and all examples 
of this phase of plant study should be noted therein. 

A few words on the proper conduct of a field trip may not 
be out of place. Foremost, as in all laboratory work, and this 
is but a phase of laboratory work, the teacher should actually 
take the trip himself a day or so previously so that all the 
illustrative material looked for may be sure to be found within 
the time allotted. The teacher must be fairly well acquainted 
with the surrounding country and should know exactly where 



180 Supervised Study in Mathematics and Science 

to go, how to get there, about the time it will take, etc. If it 
is impossible to complete the trip during a regular period then 
some other hour should be arranged, but it is better if it can 
be completed within the hour allotted. 

The discipline of a field trip will never give one any trouble, 
if the trip is well planned and if the class realize that you hold 
them as accountable for their conduct as in the regular class- 
room. Of course there will be some freedom not allowed 
ordinarily, but there should be no waste of time, no wander- 
ing from the class, and no boisterousness. The pupils are out 
for a specific purpose, and they should be made to realize that 
you will tolerate no infraction of discipline. 

The excursion should take one through the fields, the woods, 
and the pastures. Many examples of seed dispersal may be 
found, some that have already been noted in the text and some 
that perchance have been omitted. The pupils should ex- 
change experiences as they go along; and if one finds a good 
example, the others should have their attention called to it. 
Let it be more or less in the form of a game or contest to see 
who can find the most examples. Many other illustrations 
of their work in botany will also be observed, such as plant 
societies, the struggle for life, etc. 

The teacher will need to be alert every minute, and he need 
not feel abashed if the pupils ask many questions which he can- 
not answer. It would take a very wise man indeed to know 
all the secrets of nature. It is a much better sign to have 
pupils ask questions which you cannot answer than to have 
them ask no questions or show no interest in the trip. 

If seeds of plants whose names you do not know are being 
dispersed, make a record in the notebook and take back speci- 
mens to the schoolroom for later identification. Properly 



Divisions of Biology 181 

planned, the trip should be full of valuable experiences and 
should open up a new vista to the pupils. As you proceed 
in the study of this subject and later expeditions are under- 
taken, the pupils will find an endless amount of corroborative 
material for the work covered. 

Possibly the first seeds being dispersed may be the dande- 
lions'. Some late specimens of this weed may be found in the 
school yard, yet how many will ever have sensed before the 
meaning of the tiny hairlike wings of this seed? Ask the 
pupils to blow some to see how far the wind will carry them. 
Require drawings of one of the winged seeds to be made in the 
notebooks. The next specimen may be a milkweed, and as 
the pupils get into the open country they may find the 
snapdragon, beggar's-lice, etc. A cow may pass with her tail 
tangled with burdocks, a tumbleweed may blow merrily 
down the field, scattering its seeds right and left, and so on 
indefinitely may nature give an illustrated lecture of one of 
her many phases. 

LESSON VIII 
UNIT OF INSTRUCTION IX. — INSECTS 

Lesson Type. — A How to Study Lesson 

Program or Time Schedule 

The Review 10 minutes 

The Assignment 25 minutes 

The Study of the Assignment 25 minutes 

The Review. — Subject. Crustaceans. 
Method. Ask the questions found in the text, supplying 
others of a like nature, such as : 



1 82 Supervised Study in Mathematics and Science 

Name some fresh-water crustaceans. 

Name some salt-water crustaceans. 

Name some crustaceans used as food. 

What is the general weapon of defense of all crustaceans? 

Then call on someone to go to the chart and point out and 
name the various parts of the crayfish. 

The Assignment. — Tell the pupils that we are now about 
to start the study of another arthropod, the insect. Tell them 
we shall take as our type form the grasshopper, since our 
study of it will answer anatomically for all insects. Ask some- 
one to state the prominent characteristics of crustaceans. 
Then have the pupils open their books and call on someone to 
read the characteristics of insects. These are found to be : three 
body divisions and six legs. They may now be told to close 
their books again while you proceed to give them a little 
inspirational preview on insects in general. Many pupils may 
have a predetermined distaste or even horror for bugs and this 
will serve to put them in the proper attitude toward insects 
in general. Tell them that all insects are wonderfully made, 
that they have a complete, although in many insects, a very 
minute nervous system. This high-strung nervous system is 
one of the reasons why certain insects, like bees, are very 
apt to resent and resent immediately any rapid movement 
which may be interpreted by them as indicative of harm to 
them. Tell them a few of the wonderful things about ants, 
how they keep their " cows" (aphids) and milk them, how they 
organize for battle, etc. ; tell them something about the wonder- 
ful life history of the butterfly ; tell them that many scientists 
have spent their lives studying the life histories of various 
insects and have written some very interesting books about 
them, as did Romanes, Lord Avebury, Darwin, Fabre, and 



Divisions of Biology 183 

others. Tell them that some insects are very valuable to 
mankind, as the ladybug, the ichneumon fly, the honeybee ; 
and others of course are very harmful, some carrying diseases, 
as the mosquito, some destroying clothing, as the clothes moth, 
and so on. Then remark that in addition to a specific study of 
insects in general, we shall spend some time on the character- 
istics and classifications of insects, learn how to mount them, 
how to destroy the harmful ones, and how to protect the useful 
ones. In all, our study of insects is likely to be one of the 
most interesting and fascinating phases of our work in 
zoology. 

Then ask the class to name the insects they know, at the 
same time writing their names upon the board. This will 
serve to ascertain just how extensively they know insects and 
will concentrate their interest at once upon the subject, for 
they will feel that their present knowledge is of importance at 
the outset. 

The Study of the Assignment. — The study of insects and 
of the grasshopper in particular will follow closely the sugges- 
tions outlined in the study of seeds. All pupils should supply 
themselves, if possible, with live specimens of native grass- 
hoppers. If it is possible for the class to go out into the fields 
and collect some live specimens, it would be an excellent thing 
to do, but the season of the year in which this topic is reached 
may preclude this possibility, in which case the insects must be 
purchased or have been kept in cages and raised for this pur- 
pose. With the specimen before him, the pupil should ac- 
company the text with the actual handling and study of the 
insect itself. Some textbooks, as Bailey and Coleman, 1 are 
themselves guides for laboratory work and, if the teacher elects, 
1 " First Course in Biology "; The Macmillan Company, 1908. 



184 Supervised Study in Mathematics and Science 

may be made the basis for the notebook work, the pupil mak- 
ing the drawings and recording the facts learned from the 
actual observation of the grasshopper. Not all the facts 
observed should be noted therein, of course; this might be 
left to the judgment of the pupil, or the teacher might indicate 
on the board certain facts which are to be recorded. The 
notebook must never be so autocratic as to be petty ; it 
loses its essential value unless it is written up in the expectation 
that it shall serve as a check on the individual work. 

As soon as the pupils have completed the first two or three 
pages or that part of the book covering the external character- 
istics, the teacher should direct them to close their books, and 
he should review their work, asking questions corresponding 
to those of the text. This will help to emphasize the study, 
will clarify any doubtful points or rectify any mistakes in 
observation, and will further serve as an opportunity for the 
teacher to supplement with any additional data or material 
which he may feel necessary or expedient. 

In the few minutes remaining, the teacher may unfold a 
chart on the grasshopper and rapidly review the essential 
details already studied regarding the external characteristics 
of the insect. 

The Silent Study. — As far as it is possible, insist that the 
pupils find out the answers to the questions themselves. Dis- 
courage their asking you direct questions. If they make 
incorrect deductions, you can find that out later, either in the 
quiz period or from their notebooks. The pupils should be 
trained to feel that they are the ones to do the work and that 
the teacher's duty is simply to direct that work, correct wrong 
impressions, and supplement their own observations. Facts 
that the pupils ascertain for themselves will remain with 



Divisions of Biology 185 

them always, while information you give them will be but 
transitory. The teacher should always be ready to offer 
suggestions, however, to help the pupil interpret the text if 
he finds him unable to do it for himself ; but the skillful 
instructor will avoid direct answers and will strive to lead the 
pupil to form his own answer through careful questioning. 
For instance, a pupil might call the teacher to his side to ask 
him what is the general shape of the grasshopper. Instead 
of saying outright that it is cylindrical, ask him what shape it 
resembles, whether it is round or square, circular, or rectangu- 
lar, etc., whether it is uniform throughout or only suggests 
some form as a whole. If he admits that it is more or less 
irregular but that it is somewhat curved, ask him with what 
sort of curved figure it compares favorably. In other 
words, get him to make the final decision himself. If he 
is obviously wrong, try by various questions to get him to 
see that he is wrong without directly telling him so. Re- 
member that the pupils are learning judgment of fact as well 
as the facts themselves. The pupil must learn self-reliance, 
a lesson much more important than any biological fact he 
may acquire. Indeed we may safely say that all study 
should develop the powers of observation and the training 
of judgment, the ability to see and to interpret correctly 
what we see. It is one great fault with many teachers that 
they often tell too much. The teacher must know the facts, 
not to recite them but to make sure that the pupils sense 
them aright. 

If the teacher finds that several pupils are becoming con- 
fused over some question or statement, it might be well to take 
the matter up with the class as a whole and develop through 
them the correct answer or understanding, just as has been 



1 86 Supervised Study in Mathematics and Science 

suggested before for individual pupils. However, the individ- 
ual method is much to be preferred. 

The Outside Work. — Necessarily much of the work in 
biology must be done during the class period. The more this 
can be done, the better will be the results. Outside work 
should consist largely of supplementary reading, home experi- 
ments, field observations, special reports, tabulations, col- 
lecting materials, and the like. Tables of comparative study 
like that on page 85, Bailey and Coleman, give excellent out- 
side work and from their nature are quite adaptable for such 
assignments. The biology library should have a rich assort- 
ment of natural histories, bird books, flower guides, nature 
readers, etc., to which the pupils may be given assignments 
for outside reading and reports. Most textbooks furnish 
complete bibliographies, and the teacher should be reasonably 
acquainted with them. Many of the farm bulletins may be 
used, such as deal with the fly, the toad, raising bees, the 
codling moth, obnoxious weeds, etc. 

An excellent plan is to assign each day special topics for 
reports either written or oral, and to make them as varied as 
possible. They should be correlated as far as possible with 
the interest of the child, those living on farms being given some 
topic relating more or less to their environment. If some 
pupil happens to have chickens, for instance, he may be given 
a topic : " Chicken Lice and How to Eradicate Them " ; or if 
he has a pet dog, give him a pamphlet on fleas and let him 
report ; if he is the son of a physician, he may be told to look 
up and discuss the relation of the mosquito and malarial fever, 
and so on. 

Encourage the pupils to bring to class magazine or news- 
paper articles on some phase of animal life, anecdotes con- 



Divisions of Biology 187 

cerning animal intelligence, bird stories, etc. Get the pupils 
to feel that their study of biology is vital, practical, and full of 
rich interest. They will look through the papers for inter- 
esting articles, will question other people for experiences, and 
will learn to observe and interpret many of the interesting 
things that are of everyday occurrence. 

LESSON IX 

UNIT OF INSTRUCTION IX. —INSECTS 

Lesson Type. — A Laboratory Lesson 
Program or Time Schedule 

The Review 10 minutes 

The Assignment 25 minutes 

The Study of the Assignment 25 minutes 

The Review. — The few minutes of the review may be 
spent in the ordinary question method, quizzing rapidly 
various members of the class on the preceding day's work. 
The kind of questions will depend on the teacher, but they 
should at least be as thought-producing as possible, although 
the nature of the subject necessitates more or less dogmatic 
answers. The best method in asking fact questions is always 
to state the question first and assign it to some pupil after- 
wards. This intensifies the attention of all, as no one knows 
but he may be called upon. 

The Assignment. — The internal dissection of the grass- 
hopper is too complicated and minute to be required of the 
pupils. Either the teacher should make the dissection as a 
demonstration or he should use charts or blackboard drawings 
and explain the general characteristics of the anatomy from 



1 88 Supervised Study in Mathematics and Science 

them. We shall assume that the charts will be used and that 
they will resemble very closely those given in the book. 

Method. Have the pupils open their books at the paragraph 
describing the mouth parts. Have someone who is a good 
reader stand and read slowly the description of the mouth 
parts. As he proceeds, the teacher will point them out on the 
chart, interposing any additional facts or explaining the text 
in more detail as he may elect. When this has been com- 
pleted, ask some questions covering the ground studied. 

Then pass on to the next paragraph which may be that on 
respiration, and this may well be covered in the same way. In 
this way the new lesson on internal structure may be gone 
over entirely, the teacher acting as an interpreter. 

The Study of the Assignment. — I or Minimum Assign- 
ment. The three or more pages in the text covering the 
anatomy of the grasshopper. 

// or Average Assignment. Drawings, copied, of course, of 
the mouth parts, nervous system, and digestive system (in note- 
book). 

Ill or Maximum Assignment. Study of two or three micro- 
scopic slides, such as show the facets of the eye, the veining 
of the wings, and an abdominal segment containing a spiracle. 

The Silent Study. — With the assignment explained as sug- 
gested above, the pupils may not require much help during the 
study period. There may be some, however, who even now have 
not thoroughly grasped all that the author has to say, and the 
teacher will be ready to lend any additional aid if necessary. 
Particular attention should be given to the pupils who are not 
doing satisfactory work and an effort made during this period 
to aid them. This may take the method of sitting down be- 
side such a pupil and giving him personal supervision in his 



Divisions of Biology 189 

study. Let him read a few sentences and then ask him ques- 
tions on it, thus making sure that he is getting a thorough 
understanding of the meaning. Too much emphasis cannot 
be placed on the importance of the teacher's selecting each day 
someone who is f ailing behind in his work and through a study 
of his method of study trying to put him on his feet. If the 
teacher will do this at every opportunity, he will be well repaid. 
It is trite to say that the teacher should never feel satisfied 
with his work until he has exhausted every possible avenue of 
assistance to enable a pupil to master his work. Indeed it 
seems to the writer that more real good can be done in these 
individual " first aid " sections than in the regular class reci- 
tation periods. It is our business as teachers to pay partic- 
ular attention to the extremes of our classes, those doing 
minimum and those doing maximum work. 

LESSON X 
UNIT OF INSTRUCTION IX. — INSECTS 

Lesson Type. — A Correlation and Research Lesson 
Program or Time Schedule 

The Review 20 minutes 

The Assignment 15 minutes 

The Study of the Assignment 25 minutes 

The Review. — The object of this review is to emphasize 
graphically the entire order of insects as to (a) general charac- 
teristics common to all insects, (b) outstanding characteristics 
of each order, i.e. wings, (c) classification according to mouth 
parts, i.e. biting or sucking insects, and (d) economic impor- 
tance, i.e. harmful or beneficial. 



190 Supervised Study in Mathematics and Science 

Method. Write in yellow crayon, at the top of each of four 
panels of the blackboard, one of the four suggested topics. 
When the class assembles, the teacher should develop through 
questioning the data which he will write under each caption. 
When completed, the four panels should be a correlated re- 
view of the essential characteristics of insects as a class, and 
it will serve forcibly to impress upon the pupils the integrity 
of the work on the study of insects. 

Under the first caption, ask someone to name the character- 
istics which all insects have in common. If the pupil should 
mention some point which is not common to all, instead of 
ruling it out point-blank, mention some insect which does not 
possess it and thus direct the pupil how to draw upon his own 
knowledge before he answers. If the pupil should state cor- 
rectly the two characteristics desired, i.e. three pairs of legs 
and three body divisions, mention some others and ask whether 
they should be allowed. Exhaust, in other words, all possible 
items that might be included, making clear that these 
are the only ones that may be considered in this general 
summary. 

Under the next caption, write the suffix -ptera ten times, 
calling on various pupils to supply the prefix and to state the 
meaning, as diptera, two wings , coleoptera, sheath wings ; etc. 
Also call for an example under each order. 

Under the third caption, divide the board into two sections, 
one with the heading Biting Insects and the other Sucking 
Insects. Call on someone to name all those given under the 
preceding classification, which may be placed in the first group 
and those which may fall into the second group ; or possibly a 
better way would be to state each one separately and call on 
someone to tell in which group it should be placed. If a pupil 



Divisions of Biology 191 

makes a wrong classification, instead of simply declaring it 
incorrect, direct him to judge for himself whether he is right 
or wrong. A few questions concerning the characteristics 
illustrative of this order, which have been noted on the second 
board, will help him to do this. 

On the fourth board, ask first for insects which are bene- 
ficial, and insist in each case that some simple explanation be 
given to characterize its value, as Bees: make honey; Ich- 
neumon fly: destroys harmful larva of deep-boring insects through 
deposition of its eggs in their tunnels. Then under harmful 
insects, do the same. Make the questions terse, scatter them 
among the pupils, keep all on the alert for illustrations and 
for proper classifications. 

The Assignment. — As soon as the four panels have been 
completed, turn to the new work, which is a research lesson. 
Explain that you are going to give each pupil a card which will 
have written upon it a problem concerning some insect, and 
that the book or bulletin containing the information will be 
found to be indicated upon the same card. Direct them to 
study the problem, and then search for its answer in the ref- 
erences given. Tell them, when they are satisfied that they 
have found the correct answer, to write upon a sheet of paper 
the statement of the problem, its answer as they have decided 
it should be given, and then the reference by title and page. 
These papers may be collected at the close of the period or 
handed in the next day if the period does not suffice for its 
completion. 

The Study of the Assignment. — In this case there will be 
only one general assignment, but the questions may be so 
graded that the harder problems may be given to those show- 
ing more marked ability, and so on down to the easier ones 



192 Supervised Study in Mathematics and Science 

which are assigned to those who ordinarily are able to do but 
the minimum amount. 

Below is a list of suggested topics and references : 

1. When should apple trees be sprayed to kill the codling moth, 

and what is used ? See " The Control of the Codling Moth " ; 
Farmers' Bulletin No. 171. 

2. How did the cotton-boll weevil get into the United States? 

See Weed's " Farm Friends and Farm Foes" ; D. C. Heath 
and Co. 

3. What is the estimated value of a toad to the farmer? See 

" The Usefulness of the Toad" ; Farmers' Bulletin No. 196. 

4. What economic value has the cochineal bug ? See any encyclo- 

pedia. 

5. What is parthenogenesis? See Bigelow's " Applied Biology" ; 

The Macmillan Company. 

6. Why is the tachina fly beneficial? See Smallwood-Reverly- 

Bailey's " Biology for High Schools" ; Allyn and Bacon. 

7. What is Bordeaux mixture and for what is it used ? See p. 263, 

Warren's " Elements of Agriculture"; The Macmillan 
Company. 

8. How is the Rocky Mountain locust destroyed? See p. 15, 

Linville and Kelly's " Textbook in Zoology" ; Ginn and Co. 

9. Where do flies spend the winter? See p. 81, Hegner's " Prac- 

tical Zoology" ; The Macmillan Company. 
10. What is the estimated yearly economic loss from insects ? See 
Hunter's " Essentials of Biology"; American Book X!o. 

LESSON XI 

UNIT OF INSTRUCTION IX. — INSECTS 

Lesson Type. — A Socialized Lesson 
Program or Time Schedule 

The Review 25 minutes 

The Assignment 10 minutes 

The Study of the Assignment 25 minutes 



Divisions of Biology 193 

The Review. — Subject Matter. Harmful insects and how 
to get rid of them. 

Method. Previous to the assembling of the class, write upon 
the board the names of as many harmful insects as there are 
members in the class. Such a list would probably include the 
mosquito, house fly, codling moth, carpet moth, bedbug, 
cotton-boll weevil, potato bug, squash bug, tent caterpillar, 
etc. Also write upon separate slips of paper the names of the 
various members of the class, and have these in a box. 

Explain to the class that they are going to take part in a sort 
of game which might be called, " Ridding the Community of 
Obnoxious Insects.'' Each pupil will be called upon to tell 
why the insect assigned to him is harmful and how to get rid 
of it. Each time the pupil answers correctly concerning the 
insect assigned to him, the name of that insect will be erased 
from the board. The object will be to try to erase from the 
board or the community all of the names, i.e. the pests. 

Allow a few minutes for the pupils to look up in their text- 
book or any other books available any of the insects mentioned 
on the board on which they wish to be better posted. Then 
at the end of ten minutes, call someone to the front of the 
room, who might be designated class entomologist for the nonce, 
and direct him to draw some slip from the box and read the 
name upon it. The pupil whose name is read must rise and 
without help tell why this particular insect, taking it in the 
order in which it has been written upon the board, is 
harmful and how it may be destroyed. If he can recite suc- 
cessfully, the name of that insect is erased ; otherwise, it is left 
on the board, and as a penalty for failing, he may be required 
to look it up and hand in a written answer before he leaves. 
The process is repeated until all the pupils have been called 



194 Supervised Study in Mathematics and Science 

upon, and the board has been cleaned more or less completely, 
according to the successful answers. 

A little more spirit may be injected into this game if sides 
are chosen and a contest developed to see which side can 
eradicate the larger number of the harmful insects. 

The Assignment. — Tell the class that the assignment for 
the next day will be in the nature of a series of short illustrated 
lectures by the various members of the class. Tell them they 
are to bring to class some insect which they have found on 
some of their field trips, or in lieu of that, the picture of some 
insect, and that you will ask them, when called upon, to come 
forward and, holding the insect or picture in their hands, de- 
scribe to the class its characteristics , habits, order, etc. Also, 
that other members of the class may question them further 
concerning their specimens and that you will expect them to 
be posted fully on each particular insect. Suggest that they 
follow the outline below in describing their specimen : 

Name : 
Order : 
Characteristics of form : 

" " life history: 

" habitat: 
Food: 
Economic importance : 

The Study of the Assignment. — I or Minimum Assignment. 
Each pupil to be prepared to describe according to the outline 
given above, some insect to be selected by himself. 

II or Average Assignment. Each pupil to bring to class a 
specimen of the insect he is to describe, or at least a picture of it. 

III or Maximum Assignment. Be able to quote some 
source of information other than the textbook in use. 



Divisions of Biology 195 

LESSON XII 

A RED LETTER LESSON IN ZOOLOGY 

Time Schedule 

Program 60 minutes 

Purpose. — The object of a red letter lesson is to review 
vividly all the important phases of the work covered in zoology 
and also to introduce an incentive to superior work during the 
weeks devoted to the study of some particular section of the 
work in biology. It is recommended that at least one such 
lesson follow the completion of the work in botany, zoology, 
and physiology. 

Method. The program should be outlined some time in 
advance, so that all may be making their plans for this Grand 
Review. 

The program may be divided into three sections, the first or 
major part of the program being devoted to a review of the 
various orders of zoology, the other two sections of the program 
being a display of notebooks, drawings, charts, etc. Printed 
or mimeographed programs will add to the enjoyment of the 
occasion and will serve as a souvenir, something which young 
people always treasure. 

PROGRAM 

Section One: Grand Review 

A five minute talk on each of the type forms studied during the 
work in zoology : 

The Paramecium The Frog 

The Crayfish The English Sparrow 

The Grasshopper The Rabbit 

The Perch 



196 Supervised Study in Mathematics and Science 

Section Two: Display 

1. Two or three of the best notebooks. 

2. Two or three of the best written descriptions of some bird, 
in prepared booklets. 

3. A display of all the specimens which have been collected 
during the year. 

4. Charts, done in India ink, showing the fly nuisance. 

5. Bird houses, flytraps, animal snapshots taken by the pupils, 
etc. 

Section Three: Illustrated Lecture 

A stereopticon lecture, showing slides of birds common to the 
neighborhood, by the instructor. 

Procedure. If there are pupils who are especially gifted in 
the ability to draw, let them the day prior to the rendition of 
the program, make drawings of the animals suggested in 
section one, on the blackboard, using colored crayon if they 
choose. These of course may be prepared by the instructor 
if preferred, or charts may be utilized for this purpose. Every 
well-equipped biology room will necessarily be supplied with 
these charts, which should be on display at all times on the 
front wall of the room. The pupils designated to give these 
talks should be selected beforehand and they should have 
their reports well planned. 

The objects for display in section two should be selected in 
advance, and some pains taken to exhibit them in an attractive 
way. The attention of the class will have been drawn to this 
red letter lesson some time before the actual date, and the 
pupils encouraged to prepare something for it. Some of the 
charts on the fly nuisance, for instance, may easily be arranged 
with the teacher of drawing and may take the forms sug- 
gested in a little pamphlet issued by the International 



Divisions of Biology 197 

Harvester Company of New Jersey, entitled, " Trap the 
Fly." 

Pupils in the manual training department may also be en- 
couraged to make bird houses, flytraps, etc., or these may be 
borrowed from some of the pupils in the grades where this 
work is done in connection with nature study. Some very 
interesting kodak pictures of birds, mammals, and other 
animals will be forthcoming from the announcement of this 
feature. It may also serve the double purpose of interesting 
the pupils in that sport of hunting with the camera which is 
so much more to be commended than hunting with a gun. 

A short illustrated lecture on birds will also be easily ar- 
ranged if the school is supplied with a lantern. In New York 
State slides are furnished without charge by the Division of 
Visual Instruction, and in case they are not accessible from 
some free source, there are a number of firms which will rent 
or sell suitable slides very reasonably. 

LESSON XIII 
UNIT OF INSTRUCTION XVI. — BONES AND MUSCLES 

Lesson Type. — A How to Study Lesson 
Program or Time Schedule 

The Review 15 minutes 

The Assignment 20 minutes 

The Study of the Assignment 25 minutes 

The Review. — Subject Matter. Bones. 

Method. Let one of the pupils go to the skeleton or manikin 
and point to the various bones and call on individual members 
of the class to describe them as to name, kind, and use. He 



198 Supervised Study in Mathematics and Science 

is to be the judge of the correctness of the recitation given ; and 
if he accepts an erroneous answer, let him take his seat and 
assign another to act as temporary teacher. Pupils like 
occasionally to have the responsibility of playing teacher and 
it is excellent practice, calling as it does for accurate knowl- 
edge and for display of judgment. 

The Assignment. — Have some pupil step to the front of 
the room, open his book at the chapter on muscles, and read 
the first paragraph, the rest of the class meantime reading with 
him silently. Ask the one who read the paragraph if there are 
any statements of the author he does not understand. If 
there are, either bring out the meaning through careful ques- 
tioning or explain it in simple language. If there are any 
words which are new and are likely to give trouble, have some- 
one look them up in the dictionary and report. Now have 
all the pupils close their books, and ask the one who has been 
reading to state the substance of the paragraph in his own 
words. Call on one or two others to do likewise. Then pro- 
ceed in a similar manner with the next paragraph. In case 
the paragraph has necessitated quite a bit of explaining, it 
might be well to have the pupils reread it very carefully 
before reproducing it. In this way the pupils will be taught 
how they should study their new assignment. 

The principal object of this intensive study of the assignment 
will be to inculcate a method of study which shall be at once 
exhaustive, intelligent, and comprehensive. Too much so- 
called study is simply mechanical reading of the printed page 
without assimilating what the author has to say. The habit 
of retrospection of each paragraph before proceeding to the 
next is an invaluable acquisition to the pupil who desires to 
master the subject matter. 



Divisions of Biology 199 

The Study of the Assignment. — I or Minimum Assign- 
ment. 

Four or five pages of the text in the new chapter which deals 
with muscles. 

77 or Average Assignment. 

Draw diagrammatically a group of involuntary muscle 
cells. 

Ill or Maximum Assignment. 

The following or similar thought questions : 

a. Why is it important that some muscles are voluntary ? 
Name two or three. 

b. Mention some involuntary muscle which may be made 
voluntary at the desire of the owner. 

c. What makes a muscle red? tough? elastic? 

The Maximum Assignment. — When the maximum assign- 
ment consists of auxiliary thought questions, they should be 
questions that are not treated in the textbook in use, but 
should either be of such a nature as to be inferred with a little 
thought or to necessitate the use of other books. Usually the 
answers should be carefully written out and handed in the 
next day. If other books are used for authority, their titles and 
the names of the authors should be stated as references. It 
should be made emphatic that answers should be authorita- 
tive and should be substantiated by concrete references to the 
authorities. If this method becomes habituated, many loose 
statements common to-day will in time be done away with, for 
as the child learns to act and think and make statements 
during his school days when his mind is being trained to react 
along definite lines, these habits are likely to become the out- 
standing characteristic of his more mature attitude as a 
student. 



200 Supervised Study in Mathematics and Science 

LESSON XIV 

UNIT OF INSTRUCTION XVI. — MUSCLES 

Lesson Type. — A Laboratory Lesson 

Program or Time Schedule 

The Review 10 minutes 

The Assignment 10 minutes 

The Study of the Assignment 40 minutes 

The Review. — Picking out some of the important things to 
be reviewed, quiz the class rapidly and intensively on the 
things deemed essential. Many of the questions ordinarily 
asked during the recitation period may well be dispensed with 
since they are of minor importance and have presumably been 
drilled on sufficiently during the regular work on this chapter. 
This short review may be more in the light of summarizing 
some of the more essential facts about muscles. Ten minutes, 
used intensively, and thoroughly planned as to the objects 
desired, may well suffice for to-day as the laboratory work 
will take the greater part of the period. 

The Assignment. — Explain that the work to-day will be 
the study of a few microscopic slides. Have the slides that 
you wish examined carefully selected and arranged before- 
hand so that there may be no loss of time. The compound 
microscope should be adjusted and the first slide to be ex- 
amined in place. A large diagrammatic drawing of what will 
be seen under the microscope should also have been drawn 
upon the board and the attention of the class called to the 
essential things that the pupils are to try to see in the specimen. 
While the class is examining the drawing on the board, call 
someone to the microscope and have him examine the slide, 



Divisions of Biology 201 

referring occasionally to the blackboard drawing for com- 
parison and verification. Then let him return to his seat, 
copy the drawing on the board and note under it any remarks 
that seem to be necessitated by his verified examination of the 
slide. As soon as the first pupil has used the microscope, 
have another pupil ready to take his turn. In this way, with 
a little careful executive ability, the entire class will be enabled 
to examine a half dozen or more slides. 

It is here assumed that the pupil has already had experience 
with the compound microscope, but if not, then a few minutes 
should be taken at the outset to explain the workings of the 
instrument and to give directions as to how one should look 
through it. Do not allow a pupil to make any adjustments 
of the instrument as the inexperienced are not to be trusted to 
turn the adjustment screws. The slides may easily be broken, 
or the objective, and adjustment is too complicated a pro- 
ceeding for the ordinary pupil to attempt. If he finds that 
his eyes are such as to require a readjustment, the teacher 
should always give the needed assistance. At some other 
time, if the instructor sees fit, individual instruction in adjust- 
ment may be given to the pupils but always with slides made 
for the occasion and not with those forming the equipment of 
the biological laboratory. 

On Previous Arrangement of Blackboard Material. It will 
possibly have been noted in these illustrative lessons that 
much emphasis has been placed on the diagrams and data to 
be placed on the blackboard by the instructor previous to the 
assembling of the class. The science teacher above all else 
should be able to make good diagrammatic drawings. The 
use of the board should be almost wholly employed by the 
teacher. Colored crayons do much to emphasize the essential 



202 Supervised Study in Mathematics and Science 

features of the drawings, which should be carefully executed 
and labeled; they should also be fairly large. The good 
teacher is a good executive and one of the first rules of exec- 
utive ability is careful planning, with strict attention to all 
details and possible complications. 

The Study of the Assignment. — Give as a general assign- 
ment for all a study of the drawings the pupils have made and 
the verification of their authenticity through references to 
other books which give illustrations of similar drawings. Have 
them look carefully to see whether their drawings compare 
favorably with these book drawings and if not, to find out why. 

Some of the slides recommended for this lesson are : 

a. Involuntary muscle cells or fibers. 

b. Voluntary muscle cells or fibers. 

c. Heart muscle cells. 

d. Motor nerve fibers ending among fibrils of voluntary muscle. 

e. Capillaries among fibers of voluntary muscle. 

LESSON XV 
UNIT OF INSTRUCTION XVI. — MUSCLES 

Lesson Type. — A Deductive Lesson 

Program or Time Schedule 

The Review 20 minutes 

The Assignment 15 minutes 

The Study of the Assignment 25 minutes 

The Review. — Assign to as many pupils as you had differ- 
ent slides under examination the preceding day the task of 
going to the board and making from memory a rough sketch 
of the slide assigned. While they are at the board, quiz the 
class on the essential features which they found in their mi- 



Divisions of Biology 203 

croscopic study of these slides. When the drawings have been 
completed, ask the class for criticisms. If any are made which 
are well justified, have the pupil making the proper criticism 
go to the board and make the alterations. 

The Assignment. — Tell the class that the work to-day will 
be in the nature of the problem : How does muscular activity 
aid the health of the individual? The class will agree that 
muscular activity does improve one's health, but the question 
is — just how? All textbooks will have something on this 
topic, but none will deal with it exhaustively and the 
teacher will be able to make definite references to various 
authorities for more advanced research. Encourage the pupils 
to add to the references noted by discovering others. 

The Study of the Assignment. — I or Minimum Assignment. 

a. Cite cases where lack of muscular activity has resulted in 
poor health. 

Reference: p. 48, Bailey and Coleman, " First Course 
in Biology." 1 

b. Cite cases where regular muscular activity has resulted in 
improved health. 

Reference: Chapters IV and V, O'Shea and Kellogg, 
" Making the Most of Life." x 

c. Mention some muscular activities which might be good 
for improving the health. 

Reference: Chapter VII, O'Shea and Kellogg, 
" Health Habits." l 

II or Average Assignment. 

a. What becomes of muscles which are not exercised? 
Reference: Chapter II, Jewett, " The Body and Its 
Defenses." 2 

1 The Macmillan Company. * Ginn and Co. 



204 Supervised Study in Mathematics and Science 

b. What happens to muscles when they are exercised? 
Reference: Chapter III, Jewett, " The Body and Its 

Defenses." l 

c. Explain the biological result of exercise on muscles. 
Reference : Lagrange, " Physiology of Bodily Exercise." 2 

III or Maximum Assignment. 

a. Discuss the following exercises as to their specific value : 
walking, swimming, chopping wood, playing tennis, setting- 
up exercises, use of gymnastic apparatus. 

Reference : Chapter XXII, Hutchinson, " Handbook of 
Health." 3 

b. List some cautions to keep in mind while exercising. 
Reference : Chapter XIX, Eddy, " Textbook in General 

Physiology and Anatomy." 4 

c. Discuss : A healthy mind needs a healthy body. 
Reference: Lee, "Play in Education" 5 ; Groos, "The 

Play of Man." 2 

LESSON XVI 

UNIT OF INSTRUCTION XVI. — MUSCLES 

Lesson Type. — A Lesson in Correlation 

Program or Time Schedule 

The Review 20 minutes 

Program 40 minutes 

The Review. — Call on different pupils to stand and tell all 
they can to substantiate the truth of the proposition that 
muscular activity tends to good health. Let each recitation 

1 Ginn and Co. 2 D. Appleton and Co. 3 Houghton Mifflin Co. 
4 American Book Co. 5 The Macmillan Company. 



Divisions of Biology 205 

be as complete as possible, being in the nature of an exposition 
of this particular problem. At the conclusion of each recita- 
tion, make any suggestions that will tend to direct the next 
pupil to make his contribution logical and clear. Make the 
review not only scientific in nature but an exposition in good 
English, thus directly correlating the work with the oral 
English. The teacher should never accept as final any ex- 
tended recitation which does not follow the fines of correct 
English, logical sequence of thought and well-rounded sen- 
tences. Habituate the pupils in the use of short sentences, 
pure English, and definite statements. Explain that the mere 
reciting of scientific data without regard to its recital in the 
best structural form causes the loss of much of its inherent 
value. It is a pet theory of the author that aside from a 
short course in technical English, the place to teach rhetoric 
and English composition efficiently is in the various subjects 
of the school program. The teacher should feel that primarily 
he is to develop the power of the pupil to express himself and 
to sense that science, history, etc. are important largely for 
the fact that they supply the pupil with the material for his 
conversation and composition. A man may be very proficient 
in his knowledge of bird lif e, but if he is unable to express him- 
self in a clear, logical, and pleasing manner, his knowledge is 
apt to be of but little use to himself or to others. Teachers 
are too often prone to be satisfied with the mere acquisition of 
facts and to pass over as of little or slight importance the 
exposition of those facts. 

The Assignment. — The Object of a Lesson in Correlation. 
All school work should have more or less interrelationship. 
The correlation of English and the work in biology has just 
been noted. The work of muscles and their development 



206 Supervised Study in Mathematics and Science 

through athletic and other activities have been for the past few 
lessons the object of investigation. Systematic physical train- 
ing has been mentioned as one of the best means of developing 
the muscles and thus improving the health of the individual. 
Explain that you have asked the physical instructor to give 
the class a series of exercises, with proper explanations, which 
will tend to develop the various muscles of the body. Ask 
them to take notes on the various exercises and to be able to 
demonstrate each to-morrow, with a few words regarding their 
special functions. 

The physical instructor may now be introduced, and he will 
proceed to demonstrate various setting-up exercises which 
have as their essential object the strengthening of various sets 
of muscles. If the plan of the demonstration has been talked 
over with the physical instructor prior to the class period, a 
very helpful and interesting lesson will result. The physical 
instructor on his side will welcome this opportunity to demon- 
strate the scientific basis for his work, and the pupils will come 
to have a new realization of the specific value of the setting-up 
exercises which they have been doing from day to day, and 
physical training will have a new meaning for them. The 
value of the demonstration will depend, of course, upon the 
exposition of the reasons for each exercise and its hygienic and 
physiologic function. 

LESSON XVII 

AN EXAMINATION LESSON 

Part I 

{Answer all jive questions) 

1. Name four food nutrients other than water, and name a food in 
which each nutrient predominates. 



Divisions of Biology 207 

2. Describe the different kinds of teeth and their special adaptation 

for their respective functions. 

3. Compare arteries with veins as to structure and function. 

4. Name three organs of excretion, and name a waste product 

given off by each. 

5. Describe the effects of alcohol on the nervous system ; on 

digestion. 

Part II 

(Answer any three) 

6. Why should we masticate thoroughly? take systematic ex- 

ercise ? clean the teeth regularly ? drink only pure water ? 

7. Compare the human body with an engine in three particulars. 

8. Why must athletes abstain from the use of alcohol and tobacco ? 

9. How may a knowledge of biology help us to live longer? 

(Touch on at least three phases.) 

Part III 
(Answer any two; reference to library or other books allowed) 

10. Discuss some contagious disease as to source of contagion, 

symptoms, treatment, after effects. 

11. By the use of the table on food values, compute the food values 

and calories of a meal consisting of: one grapefruit, one 
boiled egg, two Vienna rolls, one pat of butter, one baked 
apple, a glass of milk, and one doughnut. 

12. Look up in some textbook on physiology other than the one 

you use, one of the following topics and report in detail in 
your own words : gross structure of the eye ; gross structure 
of the heart ; gross structure of a kidney. 

An Analysis of the Suggested Examination. The object of 
such an examination, divided into sections, is to test the pupil 
on his knowledge of certain facts, on his power to answer 
thought questions, and on his ability to look up topics in out- 
side reference books and to reproduce this knowledge in his own 



208 Supervised Study in Mathematics and Science 

words. It will be noted that the questions in Part I are 
essentially fact questions and may be said to have been selected 
from certain minimum essentials which should be required of 
all. The second part is composed of questions which will 
require the exercise of more or less thought and yet are graded 
to reach pupils of only average ability. The third group 
requires special effort on the part of the pupil and shows his 
ability to use books other than the one he has been studying. 
It tests his power of assimilating and reproducing the author's 
material and of judgment as to the selection he shall make. 

It is expected that all the pupils will answer the questions in 
the first two groups, but that only pupils of more than average 
ability and resourcefulness will attempt the last group. It is 
further planned that the grading of the paper be along these 
lines : four correct answers in group one and two in group two 
are necessary for a passing grade ; if all the answers in the first 
group and the three in the second group are correct, the grade 
will be 80; in addition, each of two correct answers in the 
third group will add ten more credits, thus giving an honor, and 
if two are correct in the third group, the result will be perfec- 
tion, or 100. 

The advantage claimed for this kind of examination papers 
is that it will allow pupils to secure a passing grade through a 
definite mastery of certain facts of minimum requirement, it 
will give added value to the grade through the ability of the 
pupils to answer correctly simple thought-producing questions, 
and will enable the candidate to secure honor marks only 
through his power to answer correctly more advanced questions 
after giving correct answers to the fact and thought questions. 
It is held that the common form of examinations, such as the 
Regents Examinations of New York State, which allow pupils 



Divisions of Biology 209 

to range from 60 to 100 in their grades, according to their 
ability to answer more or less correctly ten or more questions 
from a larger collection — all of which are of about the same 
difficulty and nature — is not a scientific method of estimating 
the pupil's knowledge or judgment. In order to receive high 
marks, the pupil should answer questions of recognized severity, 
in addition to others requiring less breadth of view but which 
are more precise in nature. Such a plan seems to encourage 
not only the mastery of certain minimum requirements but also 
advanced work during the year, for only through such extra 
research and effort may the pupil be trained to secure the cov- 
eted high grades. The old method seems to the writer too 
much like paying the artisan according to the number of times 
he can do a certain simple task well instead of according to his 
ability to do more complicated, skilled, and technical work. 
As a matter of fact the foreman in a shoe factory is not paid for 
his ability to turn out a great number of heels in a day, nec- 
essary as this is and a thing he is capable of doing, but for his 
ability to supervise the work of all the employees, manage 
men, and keep up production. This higher degree of skillful- 
ness on his part is the criterion by which he is able to secure 
higher wages ; so it should be in the examination, — the pupil 
receiving the higher wage or grade should be the pupil who in 
addition to having mastered the less technical yet important 
knowledge can prove himself capable of performing additional 
work of a higher and more advanced order. 



FIFTH SECTION 
PHYSICS 



CHAPTER SEVEN 
FURTHER LESSONS IN SCIENCE 

Space precludes any extended elaboration of illustrative 
lessons in physics, chemistry, physiography, and other ad- 
vanced sciences. It is assumed that the suggestions in biology 
will serve to illustrate possible lessons in these subjects. 
Naturally, as the pupil advances in his school life and learns 
more definitely how to study, the need of extensive directed 
study will be of less importance and necessity. For this 
reason, only a few typical lessons are given, and these in 
physics. The illustrative lessons in this subject may well 
serve, however, as suggestive of similar work in the other 
sciences. 

For the same reasons, it is deemed unnecessary to take 
space for the evaluation of the content of these subjects as has 
been done in algebra, geometry, and biology. Such an eval- 
uation of each subject into units of instruction and units of 
recitation is nevertheless important, and the teacher will do 
well to prepare such a prospectus or syllabus if he desires to 
cover the work systematically and with proportionate thor- 
oughness. 

There is a concerted effort on the part of educational author- 
ities to-day toward reorganizing the courses in all sciences in 
secondary schools along the line of the general needs of pupils 

213 



214 Supervised Study in Mathematics and Science 

and society * rather than the specialization of the content mat- 
ter. The progressive teacher will keep abreast of these new 
investigations and will evaluate his courses according to the 
latest and best suggestions. Any organization of science 
courses will therefore be more or less temporary since science is 
by nature a subject of growth and change. It is, indeed, this 
quality of progress and development which makes the study 
of science so peculiarly fascinating and vital. 

LESSON I 
UNIT OF INSTRUCTION. — FLUIDS 

Lesson Type. — An Expository and How to 
Study Lesson 

Program or Time Schedule 

The Review 10 minutes 

The Assignment 25 minutes 

The Study of the Assignment 25 minutes 

The Review. — Take up the problems on pressure of liquids, 
assigned for to-day, as follows : Call on some pupil to read the 
first example, to explain what is called for, and to tell orally 
how he solved it, giving his answer. If the answer is correct 
and all consent to his solution, pass to the next one. If there 
is some question about it, however, raised either by the teacher 
or some member of the class, direct these queries to the pupil 
reciting, making him substantiate his method or see his mistake 
if he is in error. Avoid telling him where he made his mistake 
as he will never gain the power of problem solving unless he is 
directed in the finding of his own error and also directed how 

1 " Reorganization of science in secondary schools," Bulletin No. 26, 1920, 
Federal Bureau of Education. 



Further Lessons in Science 215 

to correct the same. Each problem explained should be the 
means of clarifying and forcibly demonstrating the principles 
involved. Explaining them singly and orally concentrates 
the attention of the entire class and brings out any wrong 
impressions or false methods of solution that may have been 
practiced by anyone. It will also serve to clarify in the pupil's 
mind the process of reasoning which must mark all attempts at 
solving problems in physics. 

If the time for the review is limited as in the time schedule 
noted above, it will expedite matters to take up only those 
exercises which gave trouble. As a general thing this is the 
best way at all times, for there is little value in explaining 
problems which all have successfully solved, unless it be to 
make sure that they all solved them by correct methods. 

The Assignment. — Method. Put a fresh egg in some fresh 
water, so that the class may clearly see that the egg sinks. 
Then put it in a glass of a saturated saline solution. Ask the 
class why it does not sink now. In the same manner put a 
marble in the glass of fresh water and also in some mercury. 
Ask someone if in swimming he has ever noticed how he will 
with difficulty keep his feet if he wades out slowly into the 
water up to his mouth. Ask someone who may have lived or 
been near the ocean, if he experienced the same effect in salt 
water. These simple experiments will serve to emphasize the 
fact that bodies submerged in a liquid will be forced upward, 
the degree depending upon the nature of the liquid and the 
nature of the body submerged. Ask someone to state this 
fact in the form of a simple statement or rule. If the statement 
as given is not complete, draw out the nature of its incomplete- 
ness by skillful questioning. In other words, teach the pupils 
how to gather the results of experimentation and study and 



216 Supervised Study in Mathematics and Science 

make deductions therefrom. Then when they read in their 
textbook similar conclusions they will feel a thrill of power in 
their ability to conclude results for themselves. This idea 
should be the predominating object of the explanation of the 
assignment — to develop through the pupils' own observation 
and research the sensing of the laws and facts of physics. 

Ask if anyone knows who Archimedes was. If none knows 
anything about this great scientist, direct someone to secure 
the story of his life in an encyclopedia. While he is doing 
this, explain the meaning of the word buoyancy, using as an 
illustration the word buoy, which all will probably understand. 
Ask for illustrations of the application of buoyant force, which 
might well include the floating of logs down a river, the float- 
ing of a boat, the fact that a drowning person will come to the 
surface two or three times, etc. Referring to the report of 
the pupil concerning Archimedes and the story of his life ex- 
plain that he was the first person to discover certain laws 
which govern buoyancy. Mention the fact that we shall 
try to discover these laws for ourselves. 

Weigh some metal, as a piece of lead, in air and then weigh 
it when submerged in water. Call on some pupil to come for- 
ward and announce aloud the readings of the scales in each 
case. Ask what we understand by weight. If, then, it is the 
force of attraction between the body and the earth, ask if the 
piece of metal has really lost any weight. The pupils will 
readily see that this is impossible according to the definition. 
Ask them what has caused the apparent loss of weight. The 
class will be quick to see that there has been some readjustment 
of the weight rather than any loss. Now perform the same 
experiment, using in this case a cylindrical dish which will 
hold exactly a solid of similar form. Then if we allow the 



Further Lessons in Science 217 

weight to be submerged in water and fill the receptacle 
with the displaced water, the equilibrium will be restored. It 
will be a slow class that will not at once see that the weight 
lost in water an amount equal to the weight of the water it 
displaced. Now have someone give this in terms of a state- 
ment. After two or three have done this, each time making 
the statement more complete and clear, tell them they have 
evolved one of the laws of Archimedes. 

Referring to the previous experiment of putting the egg in 
the brine, ask them what it did in the solution. They will tell 
you it floated. Ask them what caused the egg to sink in the 
fresh water but not in the brine. Perform the experiment, 
similar to the one above, for sinking bodies. It is clearly given 
in all textbooks and need not be stated here. The principle 
for floating bodies may also be deduced as was done with the 
above principle of buoyancy. 

Ask someone to tell what is meant by mass. Stating that 

the quantity of matter or the mass in a unit volume measures 

the density or comparative density of a solid, ask if we could 

tell the density of a piece of iron, for instance, by weighing it. 

Presumably not, for it is possibly irregular in character and its 

weight, depending on its size, will vary, while its density would 

always be the same. Therefore, as in all measurements, we 

must take some unit and we shall find that the density may be 

found by dividing the mass by the volume. Put this on the 

board, first as represented by words, as 

j ., mass 

density = — , 

volume 

and then as a formula : 



218 Supervised Study in Mathematics and Science 

Weigh a piece of iron which may be measured and the volume 
of which may be computed in c.c, and then divide the one by 
the other. When this is done, ask someone to turn to the 
table of density or specific gravity in his book to compare the 
result with that given. They will be very much gratified to 
find their result tallies with that given in the book. Let some 
other pupils repeat the operation with another piece of iron ; 
then with some lead, marble, etc. 

Noting that both of these substances are regular in surface, 
ask how they would suggest proceeding with some irregular sub- 
stance, as a piece of quartz. Someone may be quick enough 
to suggest that we might get its weight by the method of water 
displacement, and then substitute in the formula as before. 

Again note that these substances are heavier than water, 
and ask someone to suggest a method for finding the specific 
gravity of something lighter than water, as paraffin. Again 
someone will probably suggest that we attach a known weight 
to the paraffin and submerge both in water, then subtract the 
known weight from the gross weight and proceed as before. 
Thus we have outlined the methods of ascertaining the specific 
gravity of solids. 

The Study of the Assignment. — I or Minimum Assignment. 
The textbook work on the subject. 

II or Average Assignment. The questions found at the end 
of the chapter covering this subject. 

III or Maximum Assignment. 

i. Tabulate in order of their density : tin, ice, gold, zinc, 
glass, and butter. 

References : any chemistry. 

2. What is the principle of the submarine? 

3. Look up about Descartes and his Cartesian diver. 



Further Lessons in Science 219 

LESSON II 

UNIT OF INSTRUCTION. — FLUIDS 

Lesson Type. — A Laboratory Lesson 
Program or Time Schedule 

The Review 10 minutes 

The Assignment 25 minutes 

The Study of the Assignment 25 minutes 

The Review. — Conduct a short, snappy recitation on the 
textbook work on density and specific gravity of solids. This 
review should serve as a preliminary review and preparation 
for the individual experiment. 

The Assignment. — During the assignment the instructor 
should outline how the experiment is to be made and what 
computations and deductions must enter into the written record. 

Method. The instructor should write on the blackboard with 
yellow crayon the object of the experiment as follows : 

Object. To compare the buoyant effect on a solid submerged 
in the water with the weight of the water which is displaced. 

Under the caption Apparatus for the present no data are to 
be placed. It is best to leave this to be filled after the actual 
experiment has been made, so that the pupil may sense the 
real object of these data, which is to be the recording of the 
apparatus used rather than some arbitrary listing of equip- 
ment. It is as premature to list the articles of apparatus be- 
fore the experiment is made as it would be to attempt to count 
the votes before an election. How many mechanics could tell 
prior to doing some repair work on an automobile just what 
tools were going to be used? 

Under Procedure, explain to the class in brief outline how 



220 Supervised Study in Mathematics and Science 

the actual experiment is going to be made, but by no means 
dictate any directions for its performance. This must be 
written up in the language of the pupils and is to follow the 
experiment. Explain that the method employed will be to 
weigh the solid in air and then in water, and that the best 
method of doing this will be to allow the scales to project over 
the table a little so that the solid may be attached to the under 
side of one scale and weighed. Then we shall take an over- 
flow can and fill it with water, holding the finger over the 
spout. Now after allowing all superfluous water to run out, 
we shall place it under the projecting solid and then allow the 
solid to drop into the bucket, catching the overflow of water in a 
catch bucket. When the body is entirely submerged, weigh the 
solid in this position. Explain that care must be taken to catch 
every drop of water and also to be as accurate as possible in 
every weighing operation. Also suggest that other methods 
of arranging the scales for this experiment may be made as the 
pupils desire, such as supporting the scales on a pile of books, 
a box, etc. 

Then ask some pupil to state what data will have been col- 
lected up to this point. Step to the board and, as these are 
stated, write them down. They will be : 

i. Weight of the solid in air. 

2. Weight of the solid submerged in water. 

3. Weight of the empty catch bucket. 

4. Weight of the catch bucket and the displaced water. 

If the four mentioned above are given in different order, 
write them down as given and afterward ask whether any would 
change the order, according to logical steps. The above order 
will probably be suggested. It will be noted by the reader 
that at every step the pupil is thrown on his own judgment as 



Further Lessons in Science 221 

to the order of procedure and not merely told these steps arbi- 
trarily. He will thus be trained to exercise his own resources 
and judgment. 

When these tabulations have been arranged for satisfactor- 
ily, ask some pupil to tell you what computations must be 
made in order to draw a conclusion. With a little skillful 
leading he mil see that we must compute from the figures 
found : (a) the loss of weight of the solid in water, and (b) the 
weight of the water displaced by the submerged solid. Using 
some fictitious numbers for data, have him tell you these two 
computations. 

Suppose for illustration, you assign these amounts to his 
list of data : 

1. Weight in air 15 gm. 

2. Weight in water 10 gm. 

3. Weight of bucket 5 gm. 

4. Weight of bucket with the displaced water . . 10 gm. 

Then our computations will be : 

1. Loss of weight of solid in water : 15 gm. — 10 gm. = 5 gm. 

2. Weight of water displaced : 10 gm. — 5 gm. = 5 gm. 

What then may we conclude from these computations? 
How do the two results compare? The pupils will of course 
see that they are the same. 

Ask someone to express this conclusion in the form of a 
statement. How does this agree with Archimedes' principle? 

Now ask someone to state the rule or formula for density. 
It is 

or density equals the mass divided by the loss of weight in 
water. 



222 Supervised Study in Mathematics and Science 

Using the data of the above, have some pupil compute the 
density of the solid, thus, 15-^5=3. 

The Study of the Assignment. — Now let each pupil get the 
apparatus he needs, make the experiment with various solids, 
and write up his experiment in the form suggested. Insist 
that each pupil get his own apparatus, and clean it up and put 
it away after the work is completed. The author has no 
sympathy with the method by which some teachers conduct a 
laboratory experiment, where all of the apparatus is set out 
before the class and blanks are distributed to the pupils, who 
participate only in the experiment by inserting the data in the 
blank spaces. This method is too much along the line of 
" press the button and we do the rest " photography. Pupils 
taught in this manner may perform experiments until dooms- 
day and they will know no more physics at the end than they 
did at the beginning. The main value of the individual labora- 
tory experiment is in the fact that the actual procedure makes 
a more lasting impression on the pupil than the mere reading 
of the experiment out of some book or witnessing a demonstra- 
ation made by the instructor. 

Drawings. When drawings may further explain just how 
the experiment was done, they are important, but when the 
process is self-evident, or easily explained, drawings are a waste 
of time. When required they should be done freehand, with 
the fewest lines possible and they should be clearly explained 
by the accompanying legend. Copied drawings are valueless 
as they do not function as they ought ; the pupil is thinking 
more of the technic of the figure than of the object for which it 
is drawn ; namely, graphically to illustrate and supplement the 
written description. 

During the performance of the experiment by the pupil, the 



Further Lessons in Science 223 

teacher should pass about to see that everything is being 
done correctly, ready with suggestions but never actually 
telling the pupil how he may better his work. In other words, 
the work of the laboratory should be used to develop the pupil's 
ability to understand the textbook, to appreciate the value of 
careful manipulation of apparatus, and to make possible the 
logical drawing of conclusions as a result of the work per- 
formed. 

LESSON III 

UNIT OF INSTRUCTION. — FLUIDS 

Lesson Type. — A How to Study Lesson 
in Problems 

Program or Time Schedule 

The Review 10 minutes 

The Assignment 25 minutes 

The Study of the Assignment 25 minutes 

The Review. — Call on some pupil to step to the front of the 
room and briefly review his entire work in the laboratory 
yesterday without referring to his notebook except for figures. 

The Assignment. — The best way to take up a set of new 
problems in physics is to have the class orally analyze a number 
of simple exercises, such as will be found in every textbook, 
thus giving a large number of pupils practice in interpreting 
what is wanted. It is also well to have a number of supple- 
mentary problems from other sources, which may be used in 
this way, if the instructor finds that the pupils do not readily 
grasp the principles involved. A large assortment of applied 
problems may be found in Lynde's Physics of the Household. 1 
1 The Macmillan Company, 191 8. 



224 Supervised Study in Mathematics and Science 

It is found that the greatest difficulty pupils have with 
problems is their inability to interpret correctly the language 
of the particular problem ; the actual mathematics is usually 
very simple. It is for this reason that the oral analysis is an 
exceedingly good method of training pupils to make the proper 
interpretation. 

It will be found expedient to have some pupil read a certain 
problem aloud ; then after giving him a few seconds for 
thought, ask him to shut his book and repeat the essential 
points of the problem. If he can state the problem in his own 
words, the instructor may feel assured that he understands 
what it is about, unless it be memorized. Now have him 
state what is to be found. With each step of his solution, 
insist on his giving a physical reason for the step. The 
teacher should never accept such bald statements as " mul- 
tiply so and so and divide the result by — ," unless the 
pupil gives as his reason for so doing some rule or physical 
fact. Also, the answer should be given in some denomina- 
tion, so that the instructor may know that the pupil has 
understood the purpose of the problem. Thus the analyses of 
the exercises become an excellent review of the principles of 
physics. 

The exercises which are assigned for the next lesson may be 
some of those taken up thus analytically in class, as well as 
some others of similar and possibly more severe character. 
These should be worked out on paper neatly and with logical 
steps, omitting of course much of the reasoning that has been 
done in the oral work. 

The Study of the Assignment. — I or Minimum Assignment. 
The first ten of the fifteen problems in the textbook. 

II or Average Assignment. The remaining five problems. 



Further Lessons in Science 225 

/// or Maximum Assignment. Problems 18-22, page 48, 
Lynde's Physics of the Household. 1 

The Silent Study. — During this part of the period the 
instructor will find that he will be needed by some of the 
weaker pupils who have still failed to master the solution of the 
problems, either through a faulty understanding of the laws 
or of the principles involved. The teacher will find this an 
opportunity for individual help and as has been emphasized 
many times throughout these lessons, he must ever be on the 
alert not to tell how to do them but to lead the pupil to solve 
them himself. 

All exercises should be collected at the close of the hour, with 
notice that the remainder will be collected to-morrow. Thus 
the teacher will know that the ones handed in are indeed the 
work of the pupils themselves, and by examining them he may 
get a clear idea of just how proficient they are becoming in 
handling this work. The exercises done outside of class are 
of doubtful value, except as they reflect the pupil's mastery of 
the principles as he later shows by his ability to do others of 
similar nature. They must be required, however, in schools 
which do not have periods lasting more than sixty minutes, 
and they should be carefully, checked and as far as possible 
carefully examined. They are of doubtful value in awarding 
grades, but the failure to hand them in or incorrect solutions 
should be reflected in the grades awarded. 

LESSON IV 

RED LETTER DAY LESSONS 

From time to time throughout the course, there should be 
provided special programs or red letter day lessons. Instead 
1 The Macmillan Company. 



226 Supervised Study in Mathematics and Science 

of limiting this lesson to any one program, a number of sug- 
gestive lessons will be mentioned, with a few words of explana- 
tion concerning each. The progressive teacher of physics 
will think of many others, of course, and he should choose the 
kinds that seem to be of most value and interest to his partic- 
ular class. 

i. A stereopticon lecture on some phase of physics. Suit- 
able slides are sold by a number of firms, as L. E. Knott 
Apparatus Co., Boston; Central Scientific Co., Chicago. 

2. If the school owns a moving picture machine, many 
valuable films may be secured from various sources. See 
Extension Leaflet No. 2, Department of the Interior, for 
list (December, 191 9). 

3. Cuts from magazines and books, photos, etc. may be used 
to advantage in giving a review of some unit of instruction 
through the use of an opaque projector. 

4. A talk to the class by some college professor of physics ; 
by the city electrician on some practical phase of his work; 
by some physician on the X-ray in surgery ; by a musician on 
the pipe organ ; etc. 

5. A trip to the electric power plant, or to the pump- 
ing station, or to a plant using a hydrostatic press, 
etc. 

6. A wireless apparatus may be set up in the laboratory and 
the hour spent in sending and receiving messages. 

7. An examination of various kinds of vacuum cleaners, 
either expository by the teacher or the actual examination of 
samples which may be collected from different sources and 
loaned for the day. Excellent cuts may be found in Lynde's 
Physics of the Household. * 

1 The Macmillan Company. 



Further Lessons in Science 227 

8. Some pupil may give an exposition of some article on 
an appropriate subject taken from a current magazine, as 
Scientific American or Popular Science Monthly. 



BIBLIOGRAPHY 

I. BOOKS FOR THE TEACHER OF MATHEMATICS 
AND SCIENCE 

Betts, G. H. — "The Recitation"; Houghton Mifflin Co., 1911. 

Cajori, Florian — "A History of Mathematics"; The Macmillan 
Company, 1919. 

Earhart, Lida B. — "Teaching Children to Study" ; Houghton Mifflin 
Co., 1909. 

Earhart, Lida B. — "Types of Teaching" ; Houghton Mifflin Co., 1915. 

Evans, G. W. — "The Teaching of High School Mathematics" ; Hough- 
ton Mifflin Co., 1911. 

Hall-Quest, A. L. — "Supervised Study"; The Macmillan Company, 
1916. 

Johnston and Others — "High School Education"; Chas. Scribner's 
Sons, 1 91 2. 

Judd, C. H. — "Psychology of High School Subjects"; Ginn and Co., 

191S. 
Lloyd and Bigelow — "The Teaching of Biology in the Secondary 

School"; Longmans, Green and Co., 1904. 
Mann, C. R. — "The Teaching of Physics" ; The Macmillan Company, 

1912. 
McMurry, F. M. — "How to Study and Teaching How to Study"; 

Houghton Mifflin Co., 1909. 
Milner, Florence — "The Teacher"; Scott, Foresman and Co., 1912. 
Parker, S. C. — " Methods of Teaching in High Schools " ; Ginn and Co., 

1915. 

Sanford, Fernando — "'How to Study, Illustrated through Physics"; 
The Macmillan Company, 1922. 

Schultze, Arthur — "The Teaching of Mathematics in Secondary 
Schools "; The Macmillan Company, 1912. 

Smith and Hall — " The Teaching of Chemistry and Physics in Second- 
ary Schools " ; Longmans, Green and Co., 1904. 

229 



230 Supervised Study in Mathematics and Science 

Smith, D. E. — "The Teaching of Elementary Mathematics"; The 
Macmillan Company, 191 7. 

Smith, D. E. — "The Teaching of Geometry'' ; Ginn and Co., 1911. 

Stoner, W. S. — "Natural Education"; Bobbs-Merrill Co., 1914. 

Strayer, G. D. — "Brief Course in the Teaching Process"; The Mac- 
millan Company, 191 2. 

Young, J. W. A. — "The Teaching of Mathematics in the Ele- 
mentary and Secondary School" Longmans, Green and Co., 191 1. 

II. MAGAZINES FOR MATHEMATICS AND SCIENCE 

TEACHERS 

The Mathematics Teacher, 41 North Queen St., Lancaster, Pa. 

School Science and Mathematics, 2059 E. 7 2d St., Chicago, 111. 

Science, The Science Press, Garrison, N. Y. 

General Science Quarterly, Salem, Mass. 

School Review, University of Chicago Press, Chicago, 111. 

Bulletins of the United States Bureau of Education, Washington, D. C. 

No. 3. Science Teaching in the Secondary Schools. 

No. 4. Mathematics in the Secondary Schools. 

No. 8. Examinations in Mathematics. 

No. 12. Training Teachers of Mathematics. 

No. 14. Report of the American Commission on Teaching of Mathe- 
matics. 

No. 16. Mathematics in Public and Private Schools. 

No. 26. Reorganization of Science in Secondary Schools. 
Monthly Record of Current Educational Publications. 

III. STANDARD TESTS AND MEASUREMENTS 

Algebra : 

Hotz's Algebra Scales, First Year; Teachers College, Columbia 

University, New York City. 
Rugg and Clark's Standardized Tests in First Year Algebra; University 

of Chicago Press, Chicago, 111. 
Thorndike's Algebra Test; Teachers College, Columbia University, 

New York City. 



Bibliography 231 

Geometry : 

Minnick's Geometry Tests; University of Pennsylvania, Philadelphia, 

Pa. 
Rogers' Mathematical Tests; Teachers College, Columbia University, 
New York City. 

Physics : 
Starch's Tests in Physics; University of Wisconsin, Madison, Wis. 



INDEX 



Absences, elimination of, 40-41 

Accuracy, 18, 65, 109, 122 

Activities, muscular, 203, 204 

Adaptations, 158, 177 

Adding machine, 30 

Addition, associative law of, 61 ; 
commutative law of, 60-61 

Agriculture, 33 

Air, fresh, 44 

Airplane, 30, 172 

Algebra, 48, 213; applications of, 36; 
bird's-eye view of course in, 33; 
Comte's definition of, 35 ; divi- 
sions of, 20-24 ; function of, 36, 54 ; 
history of, 27-28; interrelationship 
of arithmetic and, 34; methods of 
teaching intermediate, 141-145 ; 
methods of teaching advanced, 
141-145; necessity of, 31-32; ori- 
gin of the word, 27 ; practical value 
of, 29-32 ; problems as real tests 
in, 96-97 ; quotation from Milne's 
Standard, 54; representation of 
things concrete in, 75-78; Sir 
Isaac Newton's definition of, 35 ; 
solving problems in, 46 ; speed 
tests in, 73-75; "spelling-down 
bee" in, 80-81; standardized tests 
in* 74-75; technic of textbook in, 
47-48; textbook in, 51; time 
table for, 25 

Al-jebr w'al muqubalah, 27 

Allen, L. M., 10 

Amortization of interest-bearing notes, 
30 

Angles, properties of, iio-ni, 123; 



questions on, 123; theorem for 
vertical, 112, 117 

Animals, snapshots of, 196 

Answers, use of, 48; complete, 55 

Ants, 182 

Aphids, 182 

Apparatus, construction of, 152 ; physi- 
cal, 219-223; special, 151; tinker- 
ing with, 150; use of gymnastic, 
204; wireless, 226 

Aquarium, 171 

Arches, 108 

Archimedes, 216-217, 221 

Architecture, 31, 137 

Arithmetic, common errors in, 36; 
Comte's definition of, 35 ; examples 
m > 37 > 38-39; formulas used in, 
36; interdependence of algebra 
and, 37-38; nomenclature of, 37; 
pupil's present knowledge of, 60-61, 
77 ; review of fundamental processes 
in, 36-37 

Articles, magazine and newspaper, 
34, 160, 186, 226, 227 

Assignment, 6 (see also each lesson 
outlined) ; aim of, 12; average, 
15, 64, 67, 132, 151 ; completion 
of, 12-13, 64; explanation of the 
new, 133; importance of, 12; 
maximum, 15, 64, 67, 68, n 7-1 18, 
122, 132, 151, 199; minimum, 14-15, 
64, 67, 68, 129, 132, 151; nature of, 
1 2 ; study of the maximum, 121- 
122; study of, 12, 133, 162-165, 
167-168, 173-175, 177-178, 183-184, 
191-192, 202, 215-218, 222-223; 



233 



234 



Index 



summary of, 41 ; summary on the 
study of, 42 ; the threefold, 14-15, 
151; time allotted to, 12 

Assignment sheet, how to make, 14, 
63-64, 72, 151; how to use, 15-16, 
128, 130; illustration of, 19, 62; 
object of, 13-14; the threefold, 14 

Astronomy, 31 

Attention, individual, 69 

Authorities, varied opinions of, 155 

Automobile, 30, 219 

Axioms, 105 

Bacteria, 156 

Banking, 8, 182-183, 186, 191 

Bibliographies, 186 

Biennial, 171 

Biology, animal, 156; conducting a 
field trip in, 1 79-181; correlation 
of English and, 205 ; divisions of, 
I 55~ I 56; equipment of classroom 
in, 177-178, 196, 201; human, 156; 
lessons in, 213; plant, 155-156; 
problems of, 159, 166-167; survey 
of the course in, 159-160; use of 
notebooks in, 162 ; valuable lessons 
of, 157-160 

Bird houses, 196-197 

Birds, 156, 160; stereopticon lecture 
on, 196; stories about, 187; study 
of, 178 

Blackboard, use of, 16, 27, 39, 40, 41, 
42, 47, 56-57, 60, 63, 69, 71, 72, 77, 
79, 80, 82, 84, 87, 92, 94~95> I", 
113, 117, 119, 122, 123-125, 129- 
130, 134-135, 152, 158, 164, 169, 
171, 184, 187, 190-191, 193, 196, 
200-202, 202-203, 219-220; use of 
the spherical, 145 

Blood, 156 

Bonds, valuation of debenture, 30 

Bones, 156, 197 

Book, the open, 47-48 

Books, 160, 186, 193, 226; supple- 
mentary, 186 



Bordeaux mixture, 192 
Botany, 155, 195 
Bridges, 108 
Bulletins, 186 
Buoyancy, 216-217 
Burbank, Luther, 159 
Business, statistics of, 79 
Busy work, 15 
Butterfly, 182 

Cage,i57, 177, 183 

Calories, 207 

Camera, hunting with, 197 

Cancellation, 37 

Canton, N. Y., 10, 13, 68 

Capitol, at Washington, 108 

Carbon, 163, 164 

Cards, index, 143-144; problems 
written on, 191 ; reviewing Book I 
through use of, 134-135; use of 
the divided, n 4-1 16, 1 19-122 

Carelessness, 14 

Carpenter, 58 

Catcher, 86, 87 

Caterpillar, tent, 193 

Ceilings, steel, 108 

Charts, 196; making, 151, 171, 173, 
174,^ 175, 182, 184, 187, 195 

Checking, value of, 66, 245 

Chemistry, 31, 203 

Children, educating all the, 151 

Circle, 105-106, 109, 145; circum- 
ference of, 38 

Circulation, 156 

Class, testing progress of, 75 

Cochineal bug, 192 

Codling moth, 186, 192, 193 

Coefficient, 61 

Coleoptera, order of, 190 

Coloration, protective, 158 

Comte, 35 

Concentration, 44-45, 74, 108 

Cone, 109 

Contents, table of, 7, 47 

Contests, 33 



Index 



235 



Contractor, 32 

Corn, kernel of, 171, 173 

Cotton-boll weevil, 192, 193 

Course, 7; bird's-eye view of, 26, S3) 
109 

Course of study, evaluation of, 20, 
io 5> *55> 2I 3J minimum essentials, 
208 

Court, the class as a, 89-91 

Crayfish, 182, 195' 

Crayon, use of colored, 84, 190, 196, 
201, 219; waste of, 41; yellow, 71 

Credit, awarding extra, 64-65, 101 

Crustaceans, 156, 160, 181-182; ques- 
tions on, 182 

Current events, 33 

Curriculum, 6; college preparatory, 
7; domestic science, 7 

Cylinder, 109 

Dandelions, 181 

Darwin, Charles, 182 

Definitions, 21, 51, 105 

Density, 216-218; formula for, 217 

Descartes, 218 

Devices, 6, 9, 43, 71 

Diamonds, 164 

Dictionary, use of, 170 

Dietitian, 31 

Digestion, organs of, 156 

Digits, 35 

Diophantus, 28 

Diptera, order of, 190 

Discipline, 180; formal, 108 

Dividends, distribution of, 30 

Doughnuts, 88, 207 

Drawings, 109, 136, 177, 188, 195, 201, 
222 

Drill, 82 ; function of, 42 ; impor- 
tance of, 71 

Earhart, Lida, 8, 46 
Earth, size of, 107 
Education, 29, 48 



Egyptians, 28, 107, no 

Elections, 33, 79 

Embryo, 172-173, 176 

Encyclopedia, 31 

Endosperm, meaning of, 172 

Engineering, 30, 31, 107 

English, technical, 205; use of pure, 
205 

Enthusiasm, arousing, 26 

Entomologist, class, 193 

Environment, correct, 44; definition 
of, 170; importance of, 158; of the 
pupil, 178, 186; varieties of, 171 

Equation, 73-75; applications of, 
77; cubic, 29; quadratic, 20, 24, 
55, 75, 92-93; simple, 20, 23, 38; 
study of, 75-77 

Equipment, 6, 151, 196, 201 

Euclid, 107, 109, no 

Evolution, 20, 23 

Examination, a sample, 98-100, 206- 
207; an analysis of the suggested, 
207-209; value of the suggested, 
1 00-10 1 ; criticism of the ordi- 
nary, 208; final, 15; formal, 4; 
grading, 74; object of, 97; pre- 
academic, 37; regents, 3, 208-209; 
standardized tests as, 97-98; 
written, 9, 97-98, 166 

Excursions, field, 162, 194; how to 
conduct, 179-181; importance of, 
179 

Exercises, oral, 55; treatment of, 55, 
225; written, 55-57 

Exhibition, or "red letter day" lesson, 
method of conducting, 136; ob- 
ject of, 135 ; place for, 135 ; prepar- 
ation for, 135-136; program of, 

136-137 
Existence, struggle for, 169-170, 171, 

180 
Experiences, 34, 58 
Experiments, how to conduct, 174- 

I 75, 175-178; home, 178, 186 
Explanations, 40 



236 



Index 



Fabre, Henry, 158, 182 

Factoring, 8, 20, 22; exercises in, 

81-82; game of, 86-87; lesson 

on, 80-82 
Factors, highest common, 20, 22; 

modifying, 25; technical, 45 
Failures, causes of, in mathematics, 

3-4 

Federal Bureau of Entomology, 159 

Figures, rectilinear, 105, 109, 1 10-135 

Fiori, 29 

Fishes, 156, 160 

Flowers, 155, 177 

Fly, house, 193 ; tachina, 192 ; ichneu- 
mon, 183, 191 

Flytrap, 196, 197 

Foods, 156 

Forests, 155 

Formulas, algebraic, 31, 108; arith- 
metic, 36, 38, 53 

Fractions, 20, 22-23, 3^-37, 63, 75; 
complex, 83-84; definition of com- 
plex, 84; lesson on, 82-84; multi- 
plication of, 82-83 

Frog, 156, 195 

Functions, 158 

Games, ball, sSy 79> 86-87 

Geology, 31 

Geometricians, lives of, 136 

Geometry, plane, 213; applications 
of, 132; bird's-eye view of course 
in, 109; deduction of a proof in, 
1 1 2-1 14; discipline of, 108; divi- 
sions of, 105-106; history of, 107; 
meaning of the word, 107, 109; 
originals in, 127-128; practical 
value of, 107-109 ; review questions 
in, 109-110; steps taken in prov- 
ing a proposition in, 114; study of 
originals in, 124-128; suggestions 
for studying, 114-116, 118 

Geometry, solid, 141, 145-146 

Germination, 174; experiments in 
seed, 176-177 



Goethals, George W., 159 

Grade, testing for a final, 98 

Grades, arithmetic in, 37 ; supervised 

study in, 146 
Graphite, 163 
Graphs, 20, 23, $^, 75 
Grasshopper, 157-158, 182; dissection 

of the, 187, 195 
Gravity, specific, 217-218, 219 
Guidebooks, textbooks as, 161, 183 

Habitat, 158 

Hall-Quest, Alfred L., 6, 14, 63 

Hamilton, Sir William, 28 

Handwriting, 65, 135 

Health, 157-158, 203; restoration of, 

73 
Herbarium, 177 
Heron of Alexandria, 28 
Home work, n, 57-58; value of, 

58, 186-187 
Hotz' scales, 75 
House fly, 193 

Ice cream, 87-88 

Ichneumon fly, 183, 191 

Index, card, 143 ; units of, 7, 20, 47, 
51, 105, 144, 155-156, 213 

Insects, 156, 160, 182-183; a game 
about, 193-194; beneficial, 183, 
189; biting, 190; characteristics 
of, 182, 189; classification of, 183, 
189 ; economic loss from, 192 ; 
harmful, 183, 189, 193-194; in- 
teresting incidents concerning, 159, 
182; list of supplementary topics 
on, 192; pictures of, 194; sucking, 
190 

Insurance, casualty, 30 

Interest, arousing the pupils', 160; 
problems in, 38 ; theory of, 30 

Inventions, 29-30 .- 

Involution, 20, 23 > 

Iodine, 171 



Index 



237 



Iron, 163 
Italics, use of, 52 

Koch, Dr., 159 

Laboratory, supervision of work in, 
150, 219-223 

Lantern, stereopticon, 197, 226 

Leaves, 155, 167 

Legibility, importance of, 65, 135 

Lesson, aim of the, 71; definitions of 
various types of, 8-9; private, 72; 
purpose of a socialized, 88 

Lesson types : correlation, 34-43, 
189-192, 204-206; deductive, 117- 
122, 129-133, 202-204; deductive 
and how to study, 82-84, 110-116; 
examination, 95-101, 206-209; ex- 
pository and how to study, 73-80, 
92-95, 214-218; how to study, 
43-So, 123-129, 161-165, 171-175, 
181-187, 197-199, 223-225; in- 
ductive, 59-67, 67-73, 165-168, 
169-171, 175-178, 225-227; induc- 
tive and how to study, 50-59; 
laboratory, 187-189, 200-202, 219- 
223; preview, inspirational, 26- 
34, 106-110, 156-160; red letter 
day {see Program), 85-88, 135- 
*37> I 95~ I 97; socialized, 80-82, 
179-181, 192-194; socialized re- 
view, 88-91, 134-135 

Librarians, 144 

Library, contents of biologic, 186 

Life, biology, the study of, 157 

Lincoln, Abraham, 108 

Lines, parallel, 105 

Loci, 105 

Logic, 108 

McMurry, Frank M., 45 
Magazines, 34, 160, 186, 226, 227 
Mammals, 156, 160 
Man, existence of, 157; study of, 
160, 167 



Manuals, laboratory, 1 51-15 2 

Material, accumulation of, 55, 179- 
181, 186; available, 155; exami- 
nation of biologic, 161 ; importance 
of a variety of, 167 ; source of, 45, 
50, 79, 149, 160, 178-179; supple- 
mentary, 14, 45, 80, 132, 151, 165; 
treatment of, 53-55, 150, 167 

Mathematicians, pictures of, 27 

Mathematics, characteristics of, 3; 
contributors to, 28; English of, 
27; failures in, 3; language of, 52; 
mastery of, 71 ; practical value of, 
29-32 ; severity of, 4, 224 

Matter, functions of living, 158 

Memoranda, 14, 62, 130, 144 

Memorization, kinds of, 46; power 
of, 5; reliance on, 116 

Memory, employment of, 121, 224; 
function of, 46 ; overdeveloped, 4 

Mensuration, 108 

Meteorology, 31 

Method, adaptation of, 59 ; Austrian, 

37 
Microscope, use of, 188-189, 200-201 
Milkweed, 181 
Mimeograph, use of, 48, 73, 77, 127, 

151-152 

Mistakes, common, 65 ; correction 
of, 60, 65, 128-129, 132, 169; ex- 
amples of common, 38 ; glaring, 65 ; 
how to avoid, 65-66; how to find, 
84, 94, 214 

Monitors, pupils as, 92 

Monocotyledon, derivation of word, 
172 

Monomials, addition of, 21, 61, 68-70; 
division of, 22; factoring, 22, 
multiplication of, 21 ; subtraction 
of, 21 

Morale, 41, 64 

Morey, 158 

Moritz, R. E., 30-31 

Mosaics, 108 

Mosquito, 183, 186, 193 



2 3 8 



Index 



Mountains, 169 
Moving-picture machine, 226 
Multiples, common, 20, 22 
Muscles, 156, 198; biologic effect of 

exercise on, 204-206; questions on, 

199 
Museum, making a, 178 
Myers, G. W., 9 

Narcotics, 156 

Nature, dissimilarities in, 167; tran- 
quillity of, 169 
Naval Observatory, at Washington, 

31 

Navigation, 31 

Neatness, importance of, 135 

Newspapers, 34, 160, 186 

Newton, Sir Isaac, 28, 35 

New York State Education Depart- 
ment, lantern slides from, 197; 
statistics of, 3, 20; syllabus of, 20, 
155-156 

Notebooks, criticism of the work in, 
133 ; display of the best, 133, 195- 
196; how to study the returned, 
132; loose-leaf, 15, 162; manip- 
ulation of, 128-129, 130-132, 162, 
184; pupils', 125-126; record- 
ing experiments in, 164-165, 168, 
174-177, 179, 219-222; the teach- 
er's, 34; value of, 129 

Notes, historical, 34, 105 

Numbers, literal, 21 ; positive and 
negative, 20, 21, 60, 61 ; signed, 21 

Operations, fundamental, 34, 36 
Originals, rules for studying, 128; 

study of, 125-128 
Outline, for the study of insects, 194 

Panama Canal, 159 

Paraffin, 218 

Paragraph, study of the, 51-52, 54, 

188, 198; writing a, 166 
Paramecium, 195 



Parentheses, 21 ; removing of, 38, 86 

Parthenogenesis, 192 

Patterns, tile, 108 

Payments, equation of, 30 

Percentage, 8, 36 

Period, length of, 10-11, 68, 150; 
management of, 17-18 

Philosophy, school of, 108 

Phosphorus, 163, 165 

Photography, 222 

Photos, 226 

Physician, 31, 73, 186, 226 

Physics, 31, 213, 222, 223; lantern 
slides on, 226; teacher of, 226 

Physiography, 213 

Physiology, 31, 155-156, 195 

Picnic, an educational, 87-88 

Plant, electric power, 226 

Plants, cellular structure of, 155; 
classification of, 171; problems of, 
170; study of, 157, 177 

Plumule, 172 

Pointer, use of the, 124 

Polygons, areas of, 105-106; regular, 
105-106 ; similar, 105-106 

Polynomials, 68; addition of, 21, 70- 
71; division of, 22; factoring, 22; 
multiplication of, 21 ; rule for 
adding, 71; subtraction of, 21 

Postulates, 105 

Potato bug, 193 

Press, hydrostatic, 226 

Preview, inspirational, 26, 34, 106- 
110, 141, 156-160, 182; conditions 
for a successful, 33-34; in arith- 
metic, 60-61 ; method of, 26, 157- 
160; need of, 26; purpose of, 26, 
106-107, 156-157; questions on, 

35 

Principal, 136 

Prizes, 85 

Problems, analysis of several, 78 ; ap- 
plying the equation to, 77 ; business, 
30-31 ; conception of, 72 ; daily 
study of, 79, 166-167; different 



Index 



239 



kinds of, 96-97, 223; directions 
for studying, 77~78, 127, 223-224; 
questions leading up to the study 
of, 75 J recognition of, 61-62, 70- 
71; sensing, 45, 168; solving, 36; 
standardized tests on problems, 
75; statement of, 112 

Program {see Lessons, types of) : pur- 
pose of, 195; "red letter day," 
85-88, 136-137, 195-197 

Program of studies, 2 ; place of ad- 
vanced mathematics in, 141 

Proportion, 105-106 

Protractor, use of, 137 

Psychology, 31 

Pupils, characteristics of, 13, 72-73; 
classification of, 16, 18, 98 ; collec- 
tion of specimens by, 160; criti- 
cism of work by, 203 ; elimination 
of, 13, 69, 70; embarrassment of, 
72; free expression of, 34; grading 
of, 18; grouping, 72; guidance of, 
13 ; ingenuity of, 152 ; judgment of, 
82 ; judgment on part of, 135, 191 ; 
maximum, 137, 189; name of, 14; 
responsibility of, 177, 184; seating 
of, 17, 42, 72; self-reliance of, 123, 

185 
Pyramids, 107, 109 
Pythagoras, 107, no 

Quadrilaterals, 105 

Quartz, 218 

Quaternion Bridge, 29 

Questions, how to ask, 187, 191 ; im- 
portance of asking skillful, 131 

Quiz, importance of oral, 6, 166; 
method of the, 200, 202-203 

Race, championship, 85 ; division, 85 ; 

multiplication, 85 ; relay, 86 
Radicals, 20, 24, 75, 88 
Reading, necessity of intelligent, 78; 

supplementary, 151, 186 
Reasoning, faculty of, 5 ; undeveloped 

powers of, 4 



Recitation, the complete, 205; the 
unsupervised, 5 ; types of, 8 ; units 
of, 8, 21-24, 51, 105-106, 165, 213 

Record of work, 14, 16 

Recreation, 15 

Regents academic examinations, 3 

Resourcefulness, of pupils, 69, 152, 208 

Results, checking, 18, 66-67 

Review, methods of, n-12, 35, 44, 
50-51, 59-6o, 68-70, 73-75, 80-81, 
82, 8s, 88-91, 92, iio-in, 117-118, 
123-125, 129-132, 134-135, 165- 
166, 169, 171, 175, 181-182, 187, 
189-191, 192, 193-194, 197-193, 
200, 202-203, 204-205, 214-215, 
219, 223 ; nature of the, 41, 46, 161, 
175; purpose of, n; socialized, 
60, 88-91, 134-135; summary of 
the, 40-41 

Roll call, 17, 67 

Romanes, George, 159, 182 

Romans, 29, 35 

Roosevelt Dam, 108 

Roots, 155 

Rugg and Clark's tests, 74-75 

Ruler, 44 

Sanitation, 156 

Schedule, daily lesson, amplification 
of, 68; divisions of, 10-n; impor- 
tance of, 17; sample sheets, 19, 62; 
time (see Lessons, types of) 

School, the average, 122; the cor- 
respondence, 144 

Schoolroom, 9, 144 

Schultze, Arthur, 4 

Science, algebra, a general, 36; fas- 
cination of, 214; function of the 
study of, 205 ; importance of super- 
vised study in, 150, 173; popularity 
of, 149 ; practical aspect of, 149 ; 
the study of, 149 

Scorekeeper, 87 

Seeds, 155, 171-175, 176-178; dis- 
persal of j 181 



240 



Index 



Semester, 141 

Sheets, mimeograph, 48, 73, 77, 127, 
151-152 

Shipbuilding, 30 

Signs, 72 

Simpson, Mabel E., 13, 59 

Smith, Dr. Eugene, 29 

Snapdragon, 181 

Solar system, 108 

Spheres, 109 

Square, method of completing the, 93 ; 
rule for completing the, 93; the 
perfect, 86-87 

Squash bug, 193 

Standardized tests, 6, 97-98; Hotz', 
75; the Rugg and Clark's, 74, 97; 
value of, 74 

Statements, loose, 199 

Station, pumping, 226 

Statistics, 39 

Stems, 155 

Stenographer, 58 

Stereopticon, 197, 226 

Stereoscopic views, 145 

Stimulus, supplying the proper, 70 

Strayer, George D., 8 

Study period, function of, 12-13; 
management of, 17, 39, 43, 63-67; 
organization of, 46-47 ; the logical 
culmination of, 146; the teacher's 
duty during the, 38, 65-66, 118; 
work done outside of the, 57-58 

Study, how to, instruction in {see also 
how to study lessons), 43-49, 5 I- 55> 
78, 114-116, 127, 128, 131, 161-165, 
172-173, 184-198, 214-215; pur- 
pose of lessons on, 43-44, 161 

Study, cooperative, 57 ; correct habits 
of, 54; methods of, 44-46, 114-116, 
110-122, 163-165, 188-189; sum- 
mary on the silent, 43; the period 
of silent, 39-40, 63-66, 84, 94-95, 
116, 110-122, 133, 184-186, 188- 
189 ; units of, 8 ; value of outside, 
57-58,186-187 



Studying, cooperative, 57 ; rules for, 49 

Submarine, 218 

Sulphur, 163-165 

Superintendent, 136 

Supervised study, function of, 5-6, 

43-44, 69, 119, 151; Hall-Quest on, 

xiii-xvi; installation of, 10-11; 

management of, 17; meaning of, 

149; organization of, 46-47, 68; 

relationship of, 6; technic of, 6; 

value of, 5, 42, 150, 173 
Surveying, 31, 107, no, 137 
Symbols, 21, 35, 54; meaning of, 53; 

origin of, 39 
Symmetry, 106 
Sympathy, 6, 12, 13, 43, 96, 146 

Table, time, 25, 156 

Tachina fly, 192 

Tartaglia, 29 

Teacher, 205 ; activity of, 80 ; check- 
ing up work by, 66-67; duty of, 
40, 65-66, 121, 124, 131, 144, 184, 
189, 245; helps for, 63-64; judg- 
ment of, ss f 134; leadership of, 
152; opportunity of, 168; origi- 
nality and individuality of, 59; 
preparatory work of, 171-173, 179- 
180; tactfulness of, 72; the pupil 
as teacher, 198 ; use of standardized 
tests by, 74-75 

Technic, mastery of, 95, 146 

Terms, significance of, 72 ; transposi- 
tion of, 28, 76-77, 93, 94; use of, 6 

Tests, 6, 142, 166; Hotz', 75; prob- 
lems as real, 96-97; real, 95-97; 
Rugg and Clark's, 74, 97; speed, 
73-74; standardized, 6, 97-98; 
time, 45; written, 97-98 

Textbooks, 109; arithmetic, 38; 
study of, 47, 149, 183-184; sup- 
plementary, 51, 57-59, 9i, 122, 
186-199; use of, 44, 161, 203; 
varying characteristics of, 167; 
verification of, 177, 223 



Index 



241 



Thales, 30, 137 

Theorem, 112, 117; explanation of 
the new, 118; review of the, 134- 

135 

Thoroughness, importance of, 135, 
144, 165, 174 

Time {see also each lesson outlined), 
allotment of, 17, 150; amount of, 
59, 150; efficient use of, 6, 174-175 

Toad, value of, 186, 192 

Tools, 44, 152 

Triangles, 105, 109, 123; definition 
of, 118 

Trigonometry, plane, 141, 145-146 

Type forms, explanations of, 61 ; dif- 
ferent, 82 

Types, classification of exercise, 9 

Unknown, use of, 36, 76-78 



Variation, meaning of. 168 
Verification, importance of, 149 ; 

methods of, 66-67, 72~73> 93~94> 

120, 176, 202 
Vieta, 28, 35 
Volume, 217-218 

Weber-Fechner law, 31 
Weeds, 170, 186 
White, C. E., 30 
Wiley, Dr. Harvey, 159 
Wings, classification of insects ac- 
cording to, 190 
Work bench, 152 

X-ray, 226 

Zoology, 155, 183, 195 



